54  THERMIONIC  VACUUM  TUBE 

current  to  the  anode  be  observed  as  a  function  of  the  temperature 
of  the  cathode,  the  current  will  at  first  increase  until  it  reaches  a 
value  indicated  by  Ci  (Fig.  18).  Any  further  increase  in  cathode 
temperature  causes  no  further  increase  in  the  current,  and  the 
part  CiDi  of  the  curve  is  obtained.  The  current  given  by  CiDi  is 
frequently  referred  to  as  the  "  temperature  saturation  current," 
and  the  condition  characterized  by  this  lack  of  increase  of  current 
with  increase  of  cathode  temperature  as  temperature  saturation. 
The  reason  why  under  these  conditions  the  current  does  not  in- 
crease along  CiCzCz  as  would  be  expected  from  Richardson's 
equation  is  because  at  cathode  temperatures  greater  than  that  cor- 
responding to  Ci,  so  many  electrons  are  emitted  that  the  resulting 
volume  density  of  their  charge  causes  all  other  emitted  electrons 
to  be  repelled,  and  these  return  to  the  cathode.  The  applied 
voltage  EI  is  then  not  high  enough  to  draw  all  the  emitted  electrons 
away  to  the  anode.  If  now  the  voltage  be  increased  to  E2  the 
current  increases,  since  more  electrons  are  now  drawn  away  from 
the  supply  at  the  cathode,  the  full  space  charge  effect  being 
maintained  by  fewer  electrons  being  compelled  to  return  to  the 
cathode.  From  Fig.  18  it  is  seen  that  with  the  voltage  £2  the 
cathode  must  be  raised  to  a  minimum  temperature  corresponding 
to  €2  before  the  full  space  charge  effect  can  manifest  itself.  It  is 
seen,  then,  that  the  higher  the  applied  voltage,  the  higher  must 
be  the  cathode  temperature  to  obtain  the  full  space  charge  effect. 
It  is  also  seen  that  the  part  OC  of  Fig.  18  corresponds  to  the  part 
AB  of  Fig.  17,  and  CD  of  Fig.  18  to  OA  of  Fig.  17.  The  saturation 
current  is  obtained  w^hen  the  applied  voltage  is  so  high  that  a 
variation  of  voltage  does  not  cause  any  appreciable  variation  in 
current,  while  the  condition  under  which  the  thermionic  tube 
operates,  as  a  voltage  operating  device,  is  characterized  by  the 
condition  that  the  cathode  temperature  is  so  high  that  the  current 
does  not  vary  appreciably  with  variation  in  cathode  temperature. 
25.  Current-voltage  Relation  for  Infinite  Parallel  Plates.  To 
get  an  understanding  of  the  quantitative  effect  on  the  current 
by  the  space  charge  of  the  electrons,  it  may  be  well  first  to  consider 
the  ideal  and  simple  case  that  results  when  we  neglect  the  com- 
plicating factors  encountered  in  practice  and  then  consider  the 
modifications  introduced  by  these  factors.  In  deriving  the  equa- 
tions for  this  simple  oase,  we  shall  therefore  assume  that  the  elec- 
trodes are  infinitely  large  parallel  plates,  capable  of  being  main- 


PHYSICS  OF  THE  THERMIONIC  VALVE  55 

tained  at  any  desired  temperature.  Both  electrodes  will  be 
assumed  to  be  equipotential  surfaces.  The  cathode  will  be  main- 
tained at  a  high  temperature,  the  anode  remaining  cold.  It 
follows  from  Richardson's  theory  that  the  hot  plate  will  emit 
electrons,  the  emission  being  the  result  of  the  kinetic  energy  of 
the  electrons  becoming  sufficiently  high  to  overcome  the  surface 
force  that  tends  to  hold  the  electrons  within  the  cathode.  It  will 
be  recognized  that  the  energy  and  distribution  of  energy  of  the 
electrons  play  an  important  part  in  the  mechanism  of  electron 
emission.  A  derivation  of  the  relation  between  current  and 
voltage,  which  takes  into  consideration  the  energy  distribution 
between  the  electrons,  is  quite  complicated.  J.  J.  Thomson 1 
has  given  the  equations  resulting  from  the  assumption  that  the 
electrons  all  emerge  with  one  initial  velocity.  In  1911,  C.  D. 
Child  2  gave  the  full  solution,  based  on  the  assumption  that  the 
initial  velocity  of  emission  is  zero. 

Langmuir3  of  the  General  Electric  Company  and  Schottky4 
also  published  derivations  of  the  space  charge  equation  and  made 
a  careful  investigation  of  some  of  the  phenomena  observed  in 
thermionic  tubes. 

We  shall  now  derive  Child's  equation,  making  the  same  assump- 
tions, and  then  consider  the  modifications  introduced  by  a  con- 
sideration of  the  factors  neglected  in  the  simple  derivation,  and 
more  particularly  how  these  factors  contribute  to  produce  the 
type  of  current-voltage  characteristic  generally  obtained  in  prac- 
tical thermionic  tubes.  We  shall  therefore  assume  that  both 
cathode  and  anode  are  equipotential  parallel  surfaces  of  infinite 
extent,  and  that  the  electrons  emerge  from  the  hot  cathode  with 
zero  velocity.  The  cathode  C  and  anode  P  (Fig.  19)  will  be  sup- 
posed to  be  in  an  enclosure  in  which  a  perfect  vacuum  is  main- 
tained. The  degree  of  vacuum  necessary  to  approximate  this  per- 
fection will  be  discussed  in  the  next  chapter. 

The  cathode  C  can  be  raised  to  any  desired  temperature.  Let 
the  anode  P  be  raised  to  a  potential  Vi,  while  the  cathode  remains 
grounded.  As  long  as  the  temperature  of  C  is  so  low  that  practi- 
cally no  electrons  are  emitted,  the  potential  gradient  between  the 

1  J.  J.  THOMSON,  Conduction  of  Electricity  through  Gases,  2d  Ed.,  p.  223. 

2  C.  D.  CHILD,  Phys.  Rev.  Vol.  32,  p.  498,  1911. 

3 1.  LANGMUIR,  Phys.  Rev.,  (2),  Vol.  2,  p.  450,  1913. 

4  SCHOTTKY,  Jahrb.  d.  Radioaktivitat  u.  Elektronik,  Vol.  12,  p.  147,  1915. 


:  LIBRARY 


THE  THERMIONIC 
VACUUM  TUBE 

AND  ITS  APPLICATIONS 


"Ms  Qraw-MlBock  &  7m 


PUBLISHERS     OF     BOOKS 

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aiiimiiililiniiiililiiii iiiin|iiiiiiiliiiiiiliniiii  'iinifdiiiiiiiriiiiiiHliiiHiirmHihliiiiiilAniiii 


THE  THERMIONIC 
VACUUM   TUBE 

AND  ITS  APPLICATIONS 


BY 

H.  J.  VAN  DER  BIJL,  M.A.,  Ph.D. 

M.AmJ.E.E.,  M.I.R.E.,  Mem.  Am.  Phys.  Soc.,  Scientific  &  Technical 

Adviser,   Dept.  of  Mines  &  Industries,    Union  of  South  Africa, 

Late  Research  Physicist,   American   Tel.    &    Tel.   Co. 

and  Western  Electric  Co.,  New  York 


FIRST  EDITION 
THIRD  IMPRESSION 


McGRAW-HILL  BOOK  COMPANY,  INC, 

NEW  YORK:   370   SEVENTH  AVENUE 

LONDON:  6  &  8  BOUVERIE  ST.,  E.  C.  4 

1920 


Library 


COPYRIGHT,  1920  BY  THE 
McGRAW-HILL  BOOK  COMPANY,  INC. 


PREFACE 


Ix  a  comparatively  short  time  the  applications  of  Ther- 
mionics  have  grown  to  a  considerable  extent,  and  are  now  not 
only  of  great  value  in  engineering  fields,  but  are  also  penetrating 
more  and  more  into  university  and  college  laboratories.  It  is 
difficult  for  those  who  are  interested  in  the  subject,  but  who  have 
not  had  the  opportunity  or  the  time  to  follow  its  development 
closely,  to  abstract  from  the  literature,  which  has  become  quite 
voluminous,  the  principles  of  operation  of  thermionic  vacuum 
tubes.  This  and  the  popularity  which  the  remarkable  abiHty  of 
these  tubes  to  perform  a  great  variety  of  functions  has  gained  for 
them,  have  created  a  need  for  a  book  describing  in  a  connected 
manner  the  more  important  phenomena  exhibited  by  the  passage 
of  electrons  through  high  vacua. 

In  this  work  I  have  endeavored  to  set  forth  the  principles  of 
operation  of  thermionic  vacuum  tubes,  and  to  coordinate  the 
phenomena  encountered  in  a  study  of  this  field.  Such  a  proced- 
ure is  sure  to  result  in  a  more  valuable  book  than  a  detailed  descrip- 
tion without  proper  coordination  of  the  many  investigations  that 
have  been  published  on  this  subject. 

I  have  tried  to  make  the  treatment  sufficiently  elementary  to 
meet  the  demands  that  will  necessarily  be  made  on  a  book  of  this 
kind.  This  is  especially  the  case  with  the  first  few  chapters,  which 
must  be  regarded  as  very  elementary  and  are  mainly  intended 
for  those  who  are  interested  in  the  applications  of  thermionic 
tubes  but  are  not  sufficiently  acquainted  with  the  properties  and 
behavior  of  electrons  to  understand  the  operation  of  these  tubes. 

I  wish  to  express  my  indebtedness  to  several  of  my  colleagues 
who  have  read  parts  or  all  of  the  manuscript.  In  this  connection 
I  wish  to  mention  especially  Mr.  C.  A.  Richmond  and  Dr.  P.  I. 
Wold. 

H.  J.  v.  d.  B. 


CONTENTS 


INTRODUCTION xi 

CHAPTER  I 

PROPERTIES  OF  ELECTRONS 
SECTION  PAQB 

1.  Electron  and  Corpuscle 1 

2.  Lines  of  Force  and  Tubes  of  Force 2 

3.  Field  of  "Stationary  Electron." 4 

4.  Field  of  the  "Moving  Electron" 5 

5.  Mass  of  the  Electron 6 

6.  Effect  of  Electric  Field  on  the  Motion  of  an  Electron 9 

7.  Effect  of  Magnetic  Field  on  the  Motion  of  an  Electron 10 

8.  The  Accelerated  Electron.     Radiation 13 

9.  Relation  between  Space  Charge  and  Potential  Distribution 15 


CHAPTER  II 

DlSLODGMENT  OF  ELECTRONS   FROM  ATOMS  OF  VAPORS  AND   GASES. 
lONIZATION 

10.  Occurrence  of  Electrons 16 

11.  lonization 16 

12.  Constitution  of  the  Atom 17 

13.  Radiation  from  Atoms  Caused  by  Bombardment  of  Electrons 19 

14.  lonization  Voltage  and  Convergence  Frequency 21 

CHAPTER  III 

DlSLODGMENT  OF  ELECTRONS   FROM  SOLID  SUBSTANCES 

15.  Free  Electrons 23 

16.  Force  that  Holds  Electrons  in  Substances 23 

17.  Contact  Electromotive  Force 26 

18.  Measurement  of  Contact,  E.M.F .' 28 

19.  Elements  of  Thermionics. 30 

20.  Influence  of  Surface  Conditions  on  Electron  Affinity 34 

21.  Photo-electric  Effect 38 

vii 


viii  CONTENTS 

SECTION"!  PAOE 

22.  Control  of  Space  Current  by  Means  of  an  Auxiliary  or  Third  Elec- 

trode      42 

23.  Secondary  Electron  Emission.     Delta  Rays 47 


CHAPTER  IV 
PHYSICS  OF  THE  THERMIONIC  VALVE 

24.  Current-voltage  Characteristic  of  Thermionic  Valve 50 

25.  Current-voltage  Relation  of  Infinite  Parallel  Plates 54 

26.  Quantitative  Relation  for  Concentric  Cylinders 59 

27.  Influence  of  Initial  Velocities 61 

28.  Effect  of  Voltage  Drop  in  the  Filament 64 

29.  Influence  of  Limitation  of  Current  by  Thermionic  Emission •  70 

30.  Effect  of  Curvature  of  the  Characteristic 73 

31.  Energy  Dissipation  at  the  Anode 75 

32.  Efficiency  of  the  Cathode 76 

33.  Life  of  a  Vacuum  Tube 84 

CHAPTER  V 
INFLUENCE  OF  GAS  ON  THE  DISCHARGE 

34.  Volume  Effect  of  Gas.     lonization  by  Collision 86 

35.  Mean  Free  Path  of  Electrons  in  Gases 88 

36.  lonization  at  Low  Pressures 90 

37.  Effects  of  lonization  by  Collision 91 

38.  Influence  of  lonization. on  the  Infra-saturation  Part  of  the  Charac- 

teristic   93 

39.  Effect  of  Gas  on  the  Electron  Emission.     Surface  Effect 98 

40.  Influence  of  Occluded  Gases 102 

41 .  lonization  at  High  Pressures 106 

42.  Difference  between  Gas-free  Discharge  and  Arc  Discharge 107 

CHAPTER  VI 
RECTIFICATION  OF  CURRENTS  BY  THE  THERMIONIC  VALVE 

43.  Conditions  for  Rectification 109 

44.  The  Fleming  Valve Ill 

45.  Valve  Detector  with  Auxiliary  Anode  Battery 112 

46.  Thermionic  Valve  as  High  Power  Rectifier 115 

47.  Optimum  Voltage  for  Rectification 117 

48.  Types  of  Thermionic  Valves 120 

49.  Rectification  Efficiency 123 

50.  Production  of  Constant  Source  of  High  Voltage  with  the  Thermionic 

Valve 132 

51.  The  Thermionic  Valve  as  a  Voltage  Regulator .   142 


CONTENTS  ix 


CHAPTER  VII 

THE  THERMIONIC  AMPLIFIER 

SECTION  PAGE 

52.  Action  of  the  Auxiliary  Electrode 146 

53.  Current-voltage  Characteristics  of  the  Thermionic  Amplifier 150 

54.  Amplification  Constant 160 

55.  Plate  Resistance  and  Impedance 160 

56.  Mutual  Conductance 165 

57.  Shape  of  Output  Wave  in  Circuit  of  Low  External  Impedance 166 

58.  Characteristic  of  Circuit  Containing  Tube  and  Resistance  in  Series.  .  169 

59.  Static  and  Dynamic  Characteristics 170 

60.  Conditions  for  Distortionless  Amplification 178 

61.  Amplification  Equations  of  the  Thermionic  Amplifier 180 

62.  Voltage  Amplification 181 

63.  Power  Amplification 185 

64.  Experimental  Verification  of  Amplification  Equations. 189 

65.  Methods  of  Measuring  the  Amplification  Constant 193 

66.  Measurement  of  the  Plate  Resistance 195 

67.  Direct  Measurement  of  the  Mutual  Conductance 199 

68.  Circuit  for  Measuring  Amplification  Constant,  Plate  Resistance  and 

Mutual  Conductance 203 

69.  Influence  of  the  Electrode  Capacities 205 

70.  Low  Frequencies  <o  <  106 207 

71.  High  Frequencies 212 

72.  Practical  Measurement  of  Amplification 215 

73.  Amplification  as  a  Function  of  Operating  Parameters 224 

74.  Tube  Constants  as  Functions  of  the  Structural  Parameters 226 

75.  Calculation  of  Amplification  Constant .--, 227 

76.  Calculation  of  Plate  Resistance 234 

77.  Types  of  Thermionic  Amplifiers 236 

78.  Amplification  Circuits 249 

CHAPTER  VIII 
THE  VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR 

79.  Introductory 266 

80.  Method  of  Procedure  for  the  Solution  of  the  Oscillation  Equations .  .  267 

81.  Conditions  for  Oscillation  in  a  Two-element  Device 269 

82.  Conditions  for  Oscillation  for  Three-electrode  Tube 271 

83.  Relation  between  Mutual  Conductance  of  Tube  and  that  of  Plate 

Circuit 279 

84.  Phase  Relations 280 

85.  Colpitts  and  Hartley  Circuits 282 

86.  Tuned  Grid-circuit  Oscillator 284 

87.  Effect  of  Intra-electrode  Capacities — Parasitic  Circuits 285 

88.  Regeneration 286 

89.  Complex  and  Coupled  Circuits — Meissner  Circuit 290 


X  CONTENTS 

SECTION  PAGE 

90.  Circuits  Comprising  a-c.  and  d-c.  Branches 292 

91.  Effect  of  Grid  Current 295 

92.  Output  Power 296 

93.  Efficiency 298 

94.  Method  of  Adjusting  Coupling  between  Output  and  Input 306 

95.  Influence  of  the  Operating  Parameters  on  the  Behavior  of  the 

Oscillator 307 

96.  Range  of  Frequency  Obtainable  with  the  Vacuum  Tube  Oscillator 

— Circuits  for  Extreme  Frequencies 312 

CHAPTER  IX 

MODULATION  AND  DETECTION  OF  CURRENTS  WITH  THE  THERMIONIC 

TUBE 

97.  Elementary  Theory  of  Modulation  and  Detection 315 

98.  Modulation 318 

99.  Modulation  Systems 322 

100.  Detection 325 

101.  Root  Mean  Square  Values  of  Detecting  and  Modulated  Currents.. .  328 

102.  Relation  between  Detection  Coefficient  and  the  Operating  Plate 

and  Grid  Voltage 328 

103.  Detection  with  Blocking  Condenser  in  Grid  Circuit 332 

104.  Method  of  Measuring  the  Detecting  Current 335 

105.  Measurement  of  the  Detection  Coefficient 339 

106.  Detecting  Efficiency 344 

107.  Comparison  of  Detectors 346 

108.  Audibility  Method  of  Measuring  the  Detecting  Current. 349 

109.  Heterodyne  Reception  with  the  Audion 354 

110.  Zero  Beat  or  Homodyne  Method  of  Receiving  Modulated  Waves  .  .  358 

111.  The  Feed  Back  Receiving  Circuit 360 

112.  Radio  Transmitting  and  Receiving  Systems 361 

113.  Multiplex  Telegraphy  and  Telephony 364 

CHAPTER  X 
MISCELLANEOUS  APPLICATIONS  OF  THE  THERMIONIC  TUBE 

114.  The  Audion  Tube  as  an  Electrostatic  Voltmeter 367 

115.  High-tension  Voltmeter 369 

116.  The  Audion  Voltage  and  Current  Regulator 371 

117.  Power  Limiting  Devices 373 

118.  The  lonization  Manometer 375 

119.  Heterodyne  Method  of  Generating  Currents  of  Very  Low  Frequency 

with  the  Vacuum  Tube 377 

120.  The  Thermionic  Valve  as  a  High-tension  Switch 378 

121.  Devices  Employing  Secondary  Electron  Emission 378 

122.  Tubes  Containing  More  than  One  Grid 380 

Index .385 


INTRODUCTION 


THE  achievements  in  the  art  of  intelligence  communication 
which  we  have  witnessed  in  the  past  seven  or  eight  years  are  the 
result  of  an  extensive  series  of  investigations  that  stretch  over 
many  decades.  It  is  easy  to  see  the  factors  that  directly  influenced 
that  part  of  the  work  that  has  been  conducted  in  recent  years  to 
develop  our  systems  of  telephone  and  telegraph  communication. 
But  we  should  not  forget  that  this  work  rests  on  a  foundation  laid 
by  such  pioneers  as  Maxwell,  Hertz,  H.  A.  Lorentz,  J.  J.  Thomson 
and  0.  W.  Richardson — men  who  conducted  their  research  with- 
out any  monetary  motive  and  with  little  or  no  thought  to  any 
possible  future  commercial  application.  Through  their  efforts 
we  came  into  possession  of  the  electromagnetic  theory,  which  has 
taught  us  much  about  the  undulatory  propagation  of  energy, 
and  the  electron  theory,  which  enabled  us  to  explain  many  a 
baffling  phenomenon.  It  is  a  glowing  compliment  to  scientific 
research  that  just  those  two  theories  which  seemed  so  abstruse 
and  so  speculative  that  hardly  anybody  believed  in  them  should 
have  grown  into  such  valuable  commercial  assets.  The  electron 
theory  in  particular  has  been  of  invaluable  assistance  'in  the 
development  of  the  audion  or  three-electrode  thermionic  tube, 
the  device  which  forms  the  nucleus  of  the  research  and  develop- 
ment work  that  was  carried  on  in  efforts  to  improve  our  means 
of  intelligence  communication. 

The  audion  tube  consists  of  an  evacuated  vessel  containing  a 
filament  which  can  be  heated  by  passing  a  current  through  it, 
an  anode  which  is  usually  in  the  form  of  a  plate  or  pair  of  plates 
or  a  cylinder,  and  a  third  electrode,  usually  in  the  form  of  a  wire 
grating  placed  between  the  filament  and  the  anode.  The  hot 
filament  emits  electrons,  and  these  are  drawn  to  the  anode  or 
plate  under  the  influence  of  a  potential  difference  between  filament 
and  plate  such  as  to  maintain  the  plate  positive  with  respect  to  the 

xi 


xii  INTRODUCTION 

(  filament.  The  third  electrode,  commonly  referred  to  as  the  grid, 
functions  as  the  controlling  electrode  and  is  for  the  purpose  of  con- 
trolling the  flow  of  electrons  from  filament  to  anode.  By  applying 
potential  variations  to  the  grid  the  electron  current  flowing  from 
filament  to  plate  can  be  varied. 

It  has  been  known  for  many  years  that  hot  bodies  possess  the 
property  of  imparting  a  charge  to  conductors  placed  in  their 
neighborhood.  This  phenomenon  was  investigated  by  Elster  and 
Geitel  in  the  eighties  of  the  last  century.  They  found  that  if  a 
metallic  filament  was  placed  in  a  glass  vessel  and  heated  to 
incandescence  a  plate  placed  close  to  the  filament  inside  the 
vessel  acquired  a  positive  charge,  but  when  the  vessel  was  ex- 
hausted the  charge  on  the  plate  became  negative.  About  the  same 
time  Edison  noticed  that  if  a  plate  be  inserted  in  a  carbon  fila- 
ment incandescent  lamp,  a  current  flowed  between  the  plate  and 
the  filament  when  the  plate  was  connected  to  the  positive  end  of 
the  filament  but  not  when  the  plate  was  connected  to  the  negative 
end.  The  direction  in  which  the  current  appeared  to  flow  was 
from  plate  to  filament  when  the  former  was  positive  with  respect 
to  the  filament.  J.  J.  Thomson  showed  in  1899  that  the  current 
flowing  through  the  space  between  filament  and  plate  was  carried 
by  electrons.  He  did  this  by  measuring  the  ratio  of  the  charge 
to  the  mass  of  the  particles  that  appeared  to  convey  the  current, 
and  found  a  value  from  which  it  was  to  be  concluded  that  these 
particles  were  electrons.  The  mechanism  of  the  emission  of  the 
electrons  from  the  hot  filament  was  explained  by  O.  W.  Richardson 
in  1901.  Richardson  made  use  of  a  view  that  had  previously  been 
suggested  to  explain  metallic  conduction,  namely,  that  the  free 
electrons  in  a  metal  possess  kinetic  energy  like  the  molecules  of  a 
gas.  He  furthermore  assumed  that  there  is  a  force  at  the  surface 
of  the  filament  which  tends  to  hold  the  electrons  within  the  fila- 
ment, and  in  order  to  escape  from  it  the  electrons  have  to  do  a  cer- 
tain amount  of  work,  depending  on  the  value  of  this  force.  At 
ordinary  temperatures  the  energy  of  the  electrons  in  the  substance 
is  not  sufficient  to  enable  them  to  overcome  this  surface  force,  but 
if  the  temperature  of  the  filament  be  raised  sufficiently  high  the 
energy  of  the  electrons  increases  enough  to  enable  some  of  them  to 
overcome  the  surface  force  and  escape.  According  to  Richardson, 
therefore,  the  electrons  escape  from  hot  bodies  solely  in  virtue 
of  their  kinetic  energy.  A  large  number  of  experiments  have  been 


INTRODUCTION  xiii 

V 

made  by  various  investigators  on  this  phenomenon.  These 
experiments  not  only  verified  Richardson's  theory,  but  also  gave 
results  that  have  an  important  bearing  on  the  operation  of  the 
thermionic  vacuum  tube.  The  study  of  the  emission  of  electrons 
from  a  hot  filament  and  their  transport  to  the  anode  or  "  plate  " 
involves  a  large  number  of  problems,  the  solution  of  which  is 
based  on  the  electron  theory.  The  most  important  developments 
of  the  thermionic  vacuum  tube  were  carried  out  by  men  who  were 
familiar  with  this  theory,  and  indeed  their  knowledge  of  its  funda- 
mental principles  contributed  in  a  large  measure  to  the  rapidity 
with  which  the  thermionic  tube  was  nursed  from  the  stage  almost 
of  a  scientific  toy  to  a  very  important  commercial  device.  For 
these  reasons  the  first  few  chapters  of  this  book  are  devoted  to  an 
elementary  discussion  of  the  properties  of  electrons  and  the 
phenomena  encountered  in  the  conduction  of  electricity  by  dis- 
lodged charges.  The  discussion  of  these  phenomena,  however, 
has  to  be  elementary  and  concise  in  consideration  of  the  great 
amount  of  material  dealing  directly  with  the  applications  of  the 
vacuum  tube.  In  choosing  the  material  for  these  earlier  chapters 
I  was  guided  mainly  by  my  own  experiences  in  my  work  on  the 
development  of  this  type  of  device. 

When  Richardson  in  1901  gave  an  explanation  of  the  mechanism 
of  the  emission  of  electrons  from  hot  bodies  he  made  an  important 
contribution  to  science,  but  there  was  at  that  time  no  thought  of 
the  practical  value  that  this  theory  was  destined  to  have.  It  was 
in  1905  that  J.  A.  Fleming  conceived  the  idea  of  using  a  thermionic 
valve,  that  is,  an  evacuated  bulb  containing  a  cold  anode  and  a  hot 
filament  as  a  rectifier  for  the  detection  of  electromagnetic  waves. 
At  about  the  same  time  Wehnelt,  who  had  previously  carried  out 
investigations  on  the  emission  of  electrons  from  hot  bodies  and  had 
produced  the  oxide-coated  filament  which  bears  his  name,  and 
which  gives  off  electrons  much  more  readily  than  metallic  fila- 
ments, also  suggested  using  a  hot  cathode  device  as  a  rectifier  and 
described  experiments  that  he  had  conducted  with  such  a  tube. 
Since  electrons  are  emitted  from  the  hot  filament  but  not  from  the 
cold  anode,  such  a  device  has  a  unilateral  conductivity,  and  can 
therefore  be  used  to  rectify  alternating  currents.  Wehnelt's 
idea  in  using  a  hot  cathode  rectifier  was  to  obtain  large  currents 
for  small  potential  differences  between  the  electrodes.  If,  for 
example,  a  glow  discharge  is  passed  through  a  partially  evacuated 


xiv  INTRODUCTION 

tube  containing  cold  electrodes  the  space  between  the  electrodes 
contains  both  electrons  and  positive  ions.  The  electrons  move  to 
the  anode  while  the  positive  ions  move  towards  the  cathode. 
Since  the  positive  ions  are  large  and  heavy  compared  with  the 
electrons  they  move  more  slowly  than  the  electrons  under  the 
influence  of  the  same  field.  -The  space  in  the  neighborhood  of  the 
cathode  therefore  contains  more  positive  ions  than  electrons, 
and  this  causes  the  establishment  of  a  positive  space  charge  near  the 
cathode.  This  positive  space  charge  is  responsible  for  a  large 
part  of  the  voltage  drop  in  such  tubes.  By  using  a  hot  cathode 
which  spontaneously  emits  electrons  Wehnelt  could  neutralize 
the  space  charge  due  to  the  positive  ions,  because  the  electrons 
coming  from  the  cathode  combine  with  the  positive  ions  to  form 
neutral  gas  molecules.  By  this  means  therefore  the  voltage  drop 
in  the  tube  was  greatly  decreased. 

If  the  tube  be  evacuated  to  such  an  extent  that  there  are 
practically  no  collisions  between  the  electrons  and  the  residual 
gas  molecules,  the  space  between  cathode  and  anode  contains  prac- 
tically only  electrons,  and  therefore  there  is  only  a  negative  space 
charge  between  the  electrodes  of  the  tube.  We  then  have  what 
is  commonly  known  as  a  thermionic  valve.  The  characteristics 
of  this  device  are  discussed  in  Chapter  IV,  while  the  influence  of 
gases  in  the  bulb  on  the  characteristics  of  the  valve  is  dealt  with 
in  Chapter  V. 

The  tubes  discussed  in  the  following  pages  operate  under 
conditions  that  are  characterized  by  the  absence  of  gas.  To 
maintain  this  condition  it  is  necessary  to  insure  that  the  gas 
pressure  is  at  all  times  so  low  that  the  mean  free  path  of  the 
electrons  in  the  residual  gas  is  large  compared  with  the  distance 
between  the  electrodes.  This  requires  not  only  that  the  gases  in 
the  space  be  removed  to  a  sufficient  extent,  but  also  that  the  elec- 
trodes and  walls  of  the  vessel  be  sufficiently  freed  of  occluded  gases 
by  heating  these  parts  during  the  process  of  evacuation.  This 
is  necessary  because  the  bombardment  of  the  anode  by  the  elec- 
trons and  the  heating  current  in  the  filament  during  the  operation 
of  the  tube  would  result  in  a  rise  in  temperature  of  the  electrodes 
and  walls,  thus  causing  the  liberation  of  occluded  gases  and  a  con- 
sequent impairment  of  the  vacuum.  The  extent  to  which  the 
electrodes  and  walls  of  the  tube  must  be  denuded  of  gases  during 
evacuation  depends  on  the  power  dissipated  in  the  device  during 


INTRODUCTION  xv 

operation.  If  we  are  concerned  only  with  electrodes  that  can  be 
heated  by  passing  a  current  through  them  we  can  adopt  the  prac- 
tice that  lamp  manufacturers  have  been  following  for  the  last 
twenty  or  thirty  years,  viz.,  passing  a  current  through  the  elec- 
trodes to  raise  them  to  a  higher  temperature  than  they  attain  dur- 
ing operation,  and  by  baking  the  bulbs  in  ovens  to  as  high  a  tem- 
perature as  the  glass  can  stand.  But  thermionic  devices  of  the  most 
commonly  employed  types  also  contain  electrodes  (anode  and  grid) 
which  cannot  be  heated  by  passing  a  current  through  them. 
These  electrodes  can  be  heated  during  evacuation  by  electron 
bombardment;  that  is,  by  passing  a  thermionic  current  from 
cathode  to  anode  through  the  tube.  The  amount  of  this  current 
and  the  voltages  applied  should  be  higher  than  the  values  used 
when  the  tube  is  subsequently  put  in  operation.  Although  these 
tubes  operate  under  the  condition  that  gas  has  no  influence  on  the 
discharge,  the  operation  of  the  tube  will  always  be  better  under- 
stood if  the  effects  of  small  traces  of  gases  are  known.  These 
effects  are  therefore  discussed  in  Chapter  V. 

The  thermionic  valve  (by  this  we  mean  a  two-electrode  ther- 
mionic tube)  is  still  used  at  the  present  time  as  a  rectifier  of  alter- 
nating current,  and  as  such  it  is  a  valuable  instrument,  and  is 
capable  of  performing  useful  functions.  Its  operation  and  some 
of  its  uses  are  discussed  in  Chapter  VI. 

As  a  detector  of  electromagnetic  waves,  the  valve  has  no 
commercial  value.  The  device  which  is  used  for  detecting  pur- 
poses is  the  three-electrode  tube  which  in  addition  to  the  anode 
and  hot  cathode,  also  contains  a  grid  to  control  the  flow  of 
electrons  from  cathode  to  anode.  The  grid  was  inserted  by  de 
Forest  in  1907,  who  called  the  device  the  "  Audion  "  and  it  is  the 
insertion  of  the  grid  which  has  made  the  thermionic  tube  of  such 
great  value.  Since  the  flow  of  electrons  from  filament  to  anode 
or  plate  can  be  varied  by  applying  potential  variations  to  the 
grid,  the  circuit  in  which  this  tube  is  used  consists  of  two  branches : 
the  output  circuit  connecting  the  filament  to  the  plate  through  a 
current  or  power  indicating  device  and  the  input  circuit  connecting 
the  filament  to  the  grid  through  the  secondary  of  a  transformer  or 
other  means  of  supplying  potential  variations  to  the  grid. 

The  audion  was  first  used  as  a  radio  detector  but  was  sub- 
sequently found  to  be  capable  of  performing  a  number  of  other 
important  functions.  In  fact,  the  insertion  of  the  grid  into 


xvi  INTRODUCTION 

the  valve  resulted  in  a  device  of  tremendous  potentialities — 
one  that  can  justly  be  placed  in  the  same  category  with  such 
fundamental  devices  as  the  steam  engine,  the  dynamo  and  the 
telephone. 

Since  a  mere  variation  in  the  potential  of  the  grid  produces  a 
variation  in  the  plate  current,  it  could  reasonably  be  expected 
that  more  power  would  be  developed  in  the  output  circuit  than  the 
power  expended  in  the  input  circuit,  and  this  has  actually  been 
found  to  be  the  case.  The  operation  of  the  tube  as  an  amplifier 
is  simpler  than  its  operation  as  a  detector.  I  have  therefore  found 
it  best  first  to  discuss  the  manner  in  which  the  tube  operates  as  an 
amplifier,  reserving  its  operation  as  detector  for  discussion  in  a 
later  chapter.  Chapter  VII  not  only  describes  the  amplifier  and 
the  manner  in  which  it  operates  and  the  circuits  that  can  be  used, 
but  also  discusses  the  characteristics  of  the  three-electrode  tube. 

Since  the  power  in  the  output  circuit  of  the  audion  tube  is 
greater  than  the  power  expended  in  the  input,  it  is  possible  to 
increase  the  degree  of  amplification  by  feeding  back  part  of  the 
energy  in  the  output  to  the  input.  If  the  proportion  of  the 
energy  thus  returned  to  the  input  circuit  is  large  enough  and  the 
phase  relations  of  the  currents  in  the  output  and  input  circuits  are 
right,  the  tube  can  be  made  to  produce  sustained  oscillations. 
What  is  usually  done  is  to  connect  the  tube  in  an  oscillation  circuit 
having  the  desired  capacity  and  inductance,  and  then  couple  the 
output  circuit  to  the  input  in  such  a  way  that  current  variations 
in  the  output  circuit  cause  potential  variations  to  be  impressed 
on  the  grid.  There  are  a  great  variety  of  circuits  whereby  this  can 
be  accomplished.  To  make  a  three-electrode  tube  produce  sus- 
tained oscillations  is  an  extremely  simple  matter,  but  to  make  it 
operate  in  the  most  efficient  way  as  an  oscillation  generator  requires 
a  knowledge  of  the  various  factors  that  influence  its  operation 
as  such.  These  matters  are  discussed  in  Chapter  VIII. 

When  the  tube  is  used  as  an  amplifier  or  oscillation  generator 
it  is  desirable  that  the  characteristic  representing  the  relation 
between  the  plate  current  and  the  potential  of  the  plate  or  grid 
be  as  nearly  linear  as  possible.  On  account  of  the  negative  space 
charge  of  the  electrons  in  the  space  between  filament  and  anode 
the  characteristic  is  not  linear,  but  convex  towards  the  axis  of 
voltage.  When  the  applied  voltage  becomes  sufficiently  high  to 
attract  the  electrons  over  to  the  plate  as  fast  as  they  are  emitted 


INTRODUCTION  xvil 

vj 

from  the  filament,  the  characteristic  curve  becomes  concave 
towards  the  voltage  axis  and  finally  becomes  nearly  horizontal, 
thus  giving  the  saturation  current.  In  order  to  obtain  the  best 
operation  of  the  tube  as  an  amplifier,  it  is  necessary  to  straighten 
out  the  characteristic  of  the  plate  circuit.  Means  whereby  this 
can  be  done  are  discussed  in  Chapter  VII.  The  curvature  of  the 
characteristic  causes  the  shape  of  the  current  wave  in  the  output 
to  be  distorted,  and  is  therefore  a  very  undesirable  feature  of  the 
audion  as  an  amplifier.  On  the  other  hand,  the  fact  that  the 
characteristic  is  curved  simply  increases  the  number  of  uses  to 
which  the  audion  can  be  put.  Its  ability  to  detect  electromagnetic 
waves  lies  in  the  curvature  of  its  characteristic.  This  also  makes 
it  possible  to  use  the  tube  for  modulating  high  frequency  \vaves  for 
the  purpose  of  radio  or  carrier  telephony  and  telegraphy.  The 
processes  involved  in  detection  and  modulation  are  identical, 
and  these  are  therefore  treated  together  in  Chapter  IX. 

In  Chapter  X  are  described  a  few  miscellaneous  applications 
of  the  thermionic  tube.  This  list  is  intended  to  exemplify  the 
manner  of  applying  the  principles  of  the  tube  and  does  not  make 
any  pretense  at  being  complete.  It  is  believed  that  the  number  of 
such  applications  is  destined  to  increase  considerably  and  that  the 
tube  will  become  of  increasing  importance  not  only  in  engineering 
practice,  but  also  in  university  and  college  laboratories. 

A  large  number  of  names  have  been  used  to  designate  the  three- 
electrode  type  of  thermionic  tube,  such  as  audion,  pliotron, 
triode,  thermionic  valve,  etc. — an  impressive  array  of  names 
which  certainly  attests  the  importance  of  this  device.  To  fore- 
stall any  possible  confusion  in  the  mind  of  the  uninitiated  reader 
I  may  say  that  these  names  all  apply  to  one  and  the  same  thing, 
namely  the  audion  or  three-electrode  tube  discussed  in  the 
following  pages. 

The  major  portion  of  the  development  of  the  audion  has  taken 
place  in  the  past  eight  years,  but  while  the  number  of  applications 
of  the  tube  increases  almost  daily,  we  must  frankly  admit  that 
as  far  as  the  tube  itself  is  concerned,  it  was  developed  to  a  full 
grown  and  powerful  instrument  as  early  as  1914.  The  rapid 
development  of  this  device,  both  in  the  United  States  and  Europe 
and  the  popularity  it  has  gained,  are  due  to  a  number  of  factors 
that  have  concurred  to  place  it  in  the  foreground.  One  obvious 
reason,  of  course,  is  its  ability  to  perform  such  a  large  number 


xviii  INTRODUCTION 

<•  of  important  functions.      No  wonder  that  it  has  been  referred  to 
as  the  "  versatile  talking  bottle." 

Another   factor   which    stimulated   its    development    in    the 
United  States  was  the  pressing  necessity  for  a  satisfactory  system 
of   telephone  communication    over    long    distances — a  necessity 
which  resulted  from  the  recognition  of  the  telephone  as  a  very 
important  factor  in  the  industrial  and  commercial  development 
of  the  country  and  the  fact  that  the  industries  of  the  country  are 
scattered  over  extensive  regions.     It  was  evident  to  those  skilled 
in  the  art  that  the  sine  qua  non  of  such  a  system  of  telephone 
communication  is  a  device  that  will  amplify  telephone   currents 
without  impairing  the  quality  of  the  transmitted  speech  more  than 
can  be  tolerated  in  commercial  service.     When  the  audion  made 
its  appearance,  telephone  amplifiers  or  repeaters  had  already  been 
developed,   one  of  which,   the  mechanical  repeater,   still  gives 
satisfactory  service  on  some  of  the  long  distance  lines.     But  the 
potentialities  of  the  audion  were  immediately  recognized  by  the 
leading  telephone  engineers  when  it  came  into  their  hands  in  1912. 
As  the  result  of  a  far-sighted  policy  based  on  the  recognition  of  the 
influence  of  scientific  research  on  industrial  development,  the 
fate  of  this  device  was  placed  in  the  hands  of  a  number  of  well- 
trained  research  physicists  and  engineers.     Its  development  pro- 
ceeded so  rapidly  that  the  summer  of  1914  saw  the  three-electrode 
tube  as  the  repeater  on  a  commercial  system  of  telephone  com- 
munication connecting  New  York  with  San  Francisco.     In  order 
to  use  these  tubes  as  repeaters  on  such  a  long  telephone  line,  it 
stands  to  reason  that  the  tubes  must  be  carefully  designed  and 
constructed  to  have  not  only  characteristics  of  definite  prede- 
termined value,  but  characteristics  that  remain  constant  over  long 
periods  of  time  and  differ  but  little  from  one  tube  to  another. 
This  required  a  full  understanding  of  the  operation  of  the  device, 
something  which  was  secured  in  the  comparatively  short  time 
only  as  the  result  of  a  very  intensive  and  well  organized  series  of 
investigations  during  the  years  1912  to  1914.     These  investiga- 
tions were  carried  out  by  the  engineers  of  the  American  Telephone 
and  Telegraph  Company,  and  the  Western  Electric  Company. 
About  the  same  time  the  engineers  of  the  General  Electric  Com- 
pany made  a  study  of  the  characteristics  of  thermionic  tubes  and 
enriched  the  world  with  valuable  information  regarding  them. 
The  research  and  development  work  on  the  tube  during  these 


INTRODUCTION  xix 

,/ 

years  resulted  in  a  large  number  of  other  uses  to  which  it  might 

be  applied.  Its  possibilities  in  the  radio  field  were  recognized, 
and  its  application  to  this  field  resulted  in  Mai  oh,  1915,  in  the 
successful  transmission  of  speech  by  radio  from  Montauk  Point 
New  York,  to  Wilmington,  Delaware.  These  experiments  which 
were  undertaken  by  the  American  Telephone  and  Telegraph 
Company,  and  the  Western  Electric  Company,  were  continued 
with  the  cooperation  of  the  U.  S.  Navy  Department  and  resulted, 
in  the  fall  of  1915,  in  the  successful  transmission  of  speech  from 
Arlington,  Va.,  to  Paris  and  Honolulu,  a  distance  in  the  latter 
case  of  5000  miles. 

The  need  for  satisfactory  systems  of  intelligence  communica- 
tion in  the  war  zone  and  the  success  of  the  audion  tube  resulted 
in  an  industrious  development  of  this  device  and  its  applications 
in  Europe.  Most  of  the  work  that  was  done  by  the  British,  the 
French,  the  Germans,  etc.,  is  only  coming  to  light  now  after 
peace  has  been  declared.  Military  necessity  forbade  the  exchange 
of  scientific  information  that  can  reasonably  be  expected  in  times 
of  peace.  This  has  caused  a  great  deal  of  duplication  of  work, 
and  makes  it  extremely  difficult  to  give  recognition  to  individuals 
for  important  contributions.  However,  *the  matter  of  crediting 
individual  investigators  is  insignificant  in  comparison  with  the 
great  benefit  that  the  world  has  derived  from  a  general  recognition 
of  the  value  of  scientific  research  and  the  many-sided  and  intensive 
investigations  to  which  this  very  important  and  even  more  prom- 
ising device  has  been  subjected. 


THE 

THERMIONIC   VACUUM    TUBE 
AND  ITS  APPLICATIONS 


CHAPTER  I 
PROPERTIES   OF   ELECTRONS 

1.  Electron  and  Corpuscle.  The  word  electron  was  introduced 
by  Prof.  G.  Johnstone  Stoney  in  1891  to  denote  the  "  natural  unit 
of  electricity,"  that  is,  the  quantity  of  electricity  which  was 
found  to  be  invariably  carried  by  an  atom  of  any  univalent  ele- 
ment (such  as  hydrogen)  in  electrolysis.  Stoney  did  not  imply 
that  the  electron  was  a  small  particle  of  something  that  carried 
a  certain  charge — the  picture  often  formed  of  the  electron;  ac- 
cording to  his  definition  the  electron  is  simply  a  unit  of  charge 
and  that  is  all.  Corpuscle,  on  the  other  hand,  is  the  name  given 
by  Sir  J.  J.  Thomson  to  the  carriers  of  electricity  shot  off  from  the 
cathodes  in  vacuum  tubes.  The  researches  of  Thomson  on  the 
discharge  through  vacuum  tubes  showed  that  this  corpuscle  has  a 
negative  charge  equal  to  one  electron. 

The  two  terms,  electron  and  corpuscle,  are  frequently  used 
indiscriminately.  For  this  there  is  some  justification,  provided 
we  understand,  as  is  usually  done,  by  the  term  electron  the  natural 
unit  of  negative  electricity,  and  do  not  extend  its  meaning  to 
include  both  positive  and  negative  units  as  was  originally  intended 
by  Stoney.  With  this  restriction  of  the  word  electron  there  is  no 
difference  between  the  electron  and  Thomson's  corpuscle.  The 
corpuscle  was  found  to  have  a  charge  equal  to  the  electron  and  a 
mass  which  is  r§W  of  that  of  the  hydrogen  atom.  But  the  indis- 
criminate use  of  the  terms  electron  and  corpuscle  is  unfortunate, 
because  it  robs  us  of  a  name  for  the  natural  unit  of  charge,  irre- 
spective of  whether  it  is  negative  or  positive.  This  objection  is 


2  THERMIONIC  VACUUM  TUBE 

usually  overcome  by  referring  to  the  unit  of  positive  charge  as  the 
positive  electron.  This  quantity,  which  has  the  same  absolute 
value  as  the  (negative)  electron,  is  found  always  to  be  associated 
with  a  mass  about  1800  times  that  associated  with  the  electron. 
If  the  mass  of  the  electron  should  be  found  to  be  entirely  electro- 
magnetic, then  there  would  be  no  difference  between  it  and  the 
ultimate  unit  of  negative  charge. 

In  attempting  to  form  a  mental  picture  of  the  electron  it  is 
best  not  to  associate  hi  the  mind  the  idea  of  a  small  particle 
having  definite  size  and  mass,  in  the  ordinary  sense,  because  the 
size  and  mass  of  an  electron  are  things  about  which  we  must 
speak  somewhat  reservedly.  The  electron  manifests  itself  only 
in  virtue  of  the  electric  and  magnetic  fields  created  by  its  presence 
in  the  surrounding  medium,  but  whether  or  not  its  mass  is 
entirely  electromagnetic  is  as  yet  an  open  question.  Furthermore, 
the  theory  of  Abraham  and  Lorentz,  for  example,  leads  to  the  con- 
ception of  two  masses  for  the  electron,  namely,  the  so-called 
transverse  and  longitudinal  masses.  As  regards  the  size  of  the 
electron,  although  an  estimate  has  been  made  of  what  might  be 
considered  its  effective  radius  by  the  simple  process  of  integrating 
the  energy  of  the  field,  due  to  the  slow-moving  electron,  it  is  not 
unlikely  that  this  represents  only  one  size  obtained  under  one 
particular  set  of  conditions.  A  discussion  of  these  matters  is 
•beyond  the  scope  of  this  book,  and  we  must  content  ourselves 
with  only  the  most  elementary  explanation  of  some  of  the  prop- 
erties of  electrons,  and  only  insofar  as  they  are  needed  for  under- 
standing the  more  important  phenomena  encountered  in  vacuum 
tubes. 

2.  Lines  of  force  and  tubes  of  force.  When  Faraday, 
almost  a  hundred  years  ago,  found  himself  in  a  position  to  repu- 
diate the  hypothesis  of  the  action  of  forces  at  a  distance,  he  had 
formed  a  conception  of  the  nature  of  force  action  between  elec- 
trically charged  bodies  which  forms  the  basis  of  conceptions  upon 
which  modern  physics  is  built.  He  pointed  out  that  the  electrical 
energy  lies  in  the  medium  between  charged  bodies  and  not  in  the 
bodies  themselves.  This  conception,  which  is  the  parent  of  the 
electromagnetic  theory,  has  also  been  of  valuable  service  in  the 
development  of  the  electron  theory.  The  motion  of  the  electron 
through  the  medium,  the  ether,  is  somewhat  analogous  to  the 
motion  of  a  sphere  through  a  liquid;  the  moving  sphere  sets  the 


PROPERTIES  OF  ELECTRONS  3 

surrounding  liquid  in  motion  with  a  velocity  proportional  to  its 
own,  so  that  to  move  the  sphere  the  surrounding  liquid  must  also 
be  set  in  motion,  with  the  result  that  the  sphere  behaves  as  if 
its  mass  were  increased.  The  analogy  is  not  complete  because 
while  the  mass  of  the  sphere  is  apparently  increased  in  virtue  of 
its  being  surrounded  by  the  liquid,  the  mass  of  the  electron  is 
apparently  entirely  due  to  the  condition  of  the  surrounding 
medium.  The  analogy  would  be  more  nearly  complete  if  the 
sphere  were  supposed  to  be  practically  weightless,  such  as  a  tennis 
ball  kept  completely  immersed  in  a  liquid,  say  by  means  of  a  string 
tied  to  the  bottom  of  the  liquid  container.  If  we  imagine  that  the 
string  could  move  freely  parallel  to  itself,  then  the  inertia  of  the 
ball  when  moved  in  a  plane  perpendicular  to  the  direction  of  the 
string  will  be  due  almost  entirely  to  the  motion  of  the  liquid 
surrounding  the  ball.  Such  is  the  case  when  an  electron  moves 
in  a  plane.  There  is,  however,  this  important  difference:  the 
moving  electron  does  not  drag  the  ether  (that  is,  the  surrounding 
medium)  with  it,  as  the  ball  drags  the  liquid  in  which  it  is  im- 
mersed; what  is  dragged  along  is  a  modification  of  the  ether. 
To  get  an  understanding  of  this  it  is  necessary  'to  introduce  the 
conception  of  lines  and  tubes  of  force. 

The  conception  of  lines  of  force  was  introduced  by  Faraday 
to  form  a  mental  picture  of  the  processes  going  on  in  the  electric 
field.  To  him  these  lines  were  not  mere  mathematical  abstrac- 
tions. He  ascribed  to  them  properties  that  gave  them  a  real 
physical  significance.  They  terminate  on  opposite  charges,  are 
always  in  a  state  of  tension,  tending  to  shorten  themselves,  and  are 
mutually  repellent.  The  direction  of  a  line  of  force  at  any  point 
gives  the  direction  of  the  field  at  that  point.  With  the  help  of  these 
properties  of  lines  of  force  it  is  possible  to  obtain  an  idea  of  the  dis- 
tribution of  the  intensity  of  the  field  surrounding  electrically 
charged  bodies. 

The  idea  of  tubes  of  force  has  been  introduced  to  make  the 
method  of  Faraday  metrical  rather  than  merely  descriptive.  A 
tube  of  force  is  obtained  by  drawing  a  number  of  lines  of  force 
through  the  boundary  of  any  small  closed  curve.  The  lines  then 
form  a  tubular  surface  which,  it  can  be  proved,  will  never  be  cut 
by  any  lines  of  force,  and  the  extremities  of  which  enclose  equal 
and  opposite  charges.  By  properly  choosing  the  area  of  the  sur- 
face enclosed  by  the  curve  through  which  the  lines  are  drawn  the 


4  THERMIONIC  VACUUM  TUBE 

extremities  of  the  tube  can  be  made  to  enclose  unit  charge.  Such 
a  unit  tube  is  called  a  Faraday  tube.  Maxwell  and  J.  J.  Thomson 
have  made  an  exhaustive  study  of  these  tubes  of  force  and  expressed 
their  properties  in  mathematical  terms.  The  result  that  interests 
us  here  is  that  a  tube  of  force  behaves  as  though  it  had  inertia, 
so  that  in  order  to  move  a  tube  work  must  be  done.  This  explains 
why  a  charge  behaves  as  if  it  had  mass.1 

3.  Field  of  the  "  Stationary  Electron."  The  electric  field 
surrounding  a  point  charge  equal  to  one  electron  far  removed  from 
other  charges  may  be  represented  by  lines  of  force  in  the  manner 
shown  in  Fig,  1,  If  this  charge  is  at  rest  the  field  is  a  symmetrical 


FIG.  1 

electrostatic  field,  the  lines  of  force  being  distributed  uniformly 
in  radial  fashion.  This  is  what  may  be  called  the  field  of  the 
stationary  electron.  Whether  the  electron  actually  consists  of  a 
small  charge  located  at  0,  or  whether  the  modification  of  the  ether, 
as  represented  by  Fig.  1,  itself  constitutes  what  is  known  as  the 
electron,  is  as  yet  an  unanswered  question.  For  our  present 
purposes  it  makes  no  difference  which  of  the  two  views  is  the  cor- 
rect one,  for  the  following  reasons:  It  is  known  from  elementary 
electrostatics  that  a  charge  which  is  uniformly  distributed  over  the 

1  It  must  be  remarked  that  the  conception  of  tubes  of  forces  is  used  here 
merely  to  aid  in  understanding  the  phenomena.  Whether  or  not  tubes  of 
force,  or  even  the  ether,  possess  any  physical  significance  is  a  question. 
Modern  developments  seem  to  indicate  that  this  question  must  be  answered 
in  the  negative. 


PROPERTIES  OF  ELECTRONS  5 

surface  of  a  sphere  acts  as  if  the  whole  charge  were  concentrated 
at  the  center  of  the  sphere.  Hence,  if  the  isolated  electron  is  only 
a  symmetrical  radial  field  with  lines  of  force  converging  uniformly 
to  a  mathematical  point,  we  can  still  look  upon  the  field  as  being 
due  to  a  charge  concentrated  at  the  point,  or  uniformly  distributed 
over  the  surface  of  a  small  sphere  whose  center  is  at  the  point. 
We  can  therefore  treat  the  electron  as  though  it  had  definite 
size.  When  we  say  that  the  electronic  charge  is  e  we  mean 
that  the  electron  constitutes  an  electric  field  equivalent  to  that 
which  would  be  obtained  if  a  charge  equal  to  e  were  concentrated 
at  the  center  of  the  field.  The  intensity  E  of  the  field  at  a  dis- 

p 

tance  r  from  the  center  of  the  field  is  then  -5.     The  number  of 

r2 

Faraday  tubes  passing  through  unit  area  at  right  angles  to  the 
direction  of  the  tubes  can  be  expressed  by 


This  quantity  is  often  referred  to  as  the  electric  displacement. 
Maxwell's  displacement  current  is  nothing  else  but  the  time  varia- 
tion of  the  density  of  the  Faraday  tubes. 

4.  Field  of  the  "Moving  Electron."  Whenever  an  electron  is  in 
motion  it  is  accompanied  by  a  magnetic  field.  This  field  is 
created  by  the  motion  of  the  Faraday  tubes  invariably  associated 
with  the  electron.  J.  J.  Thomson  has  shown  that  if  6  be  the  angle 
between  the  direction  of  a  Faraday  tube  and  the  direction  in  which 
it  is  moving  at  a  point  P  with  a  velocity  v,  the  motion  of  the  tube 
produces  a  magnetic  force  at  P  equal  to  4.TTV  sin  6.  The  direction 
of  the  magnetic  force  is  at  right  angles  to  the  tube  and  the  direc- 
tion in  which  it  is  moving.  Hence,  if  a  charge  at  0  (Fig.  2)  were 
moving  in  the  direction  OQ  the  lines  of  magnetic  force  would  be 
circles  having  their  centers  along  OQ.  Here  we  have  a  magnetic 
field  produced  by  the  motion  of  a  charge.  But  an  electric  current 
always  produces  a  magnetic  field.  Hence  a  moving  charge  and  a 
current  have  the  same  effect.  It  can  be  shown  that  a  charge  e 
moving  with  a  velocity  v  is  equivalent  to  a  current  of  magnitude 
ve,  and  if  there  are  n  charges  the  current  is  nev.  This  result, 
though  seemingly  obvious,  is  very  important.  We  shall  have 
occasion  to  make  use  of  this  result  in  the  study  of  the  discharge 
through  vacuum  tubes.  A  circuit  in  which  a  vacuum  tube  is 


6  THERMIONIC  VACUUM  TUBE 

inserted  consists  of  two  parts:  one,  the  ordinary  metallic  circuit, 
and  the  other,  the  space  between  the  cathode  and  the  anode 
in  the  tube.  In  the  tube  we  have  to  deal  with  the  motion  of 
electrons  through  space;  this  constitutes  the  so-called  "  space 
current."  Problems  encountered  in  this  type  of  circuit  are  not  so 
easily  handled  as  those  dealing  with  ordinary  metallic  circuits, 
because  explanations  of  electrical  phenomena  in  the  latter  are 
based  on  Ohm's  law,  whereas  circuits  in  which  vacuum  tubes  are 
included  do  not  obey  Ohm's  law.  The  extent  of  the  deviation 
from  Ohm's  law  in  such  circuits  depends  upon  .the  nature  of  the 
discharge  through  the  tube  and  the  relative  values  of  the  impe- 
dance of  the  tube  and  that  of  the  metallic  circuit. 


FIG.  2. 

5.  Mass  of  the  Electron.  Let  us  consider  the  case  of  an 
electron  moving  with  uniform  speed  through  the  ether  in  the 
direction  OQ  (Fig.  2).  If  the  velocity  with  which  the  electron 
moves  is  small  compared  with  that  of  light  the  tubes  of  force  will 
be  uniformly  distributed  as  in  the  case  of  the  stationary  electron. 
Hence  the  displacement  or  density  of  the  Faraday  tubes  at  P  is, 

as  we  have  seen  above,  equal  to  ^  —  %>  and  therefore  the  magnetic 
force  produced  at  the  point  P  by  the  moving  electron  is 

ev  sin  6 


„ 
H 


It  is  known  from  elementary  physics  that  the  energy  in  unit  volume 
of  a  magnetic  field  at  a  point  where  the  magnetic  intensity  is  H 

is  -g-.     Hence  the  energy  of  the  field  at  the  point  P  is 

eV  sin2  6 
Srrr4      ' 


PROPERTIES  OF  ELECTRONS  7 

Integrating  this  over  the  whole  space  from  infinity  up  to  a  small 

«  o    o 

distance  a  from  0,  the  total  energy  of  the  field  is  found  to  be  —  '. 

3a 

Now,  the  kinetic  energy  of  mass  m  moving  with  a  velocity  v  is 
%mv2.     If  the  body  has  a  charge  e  we  have  to  add  to  this  energy 


-5—,  so  that  the  total  energy  of  the  system  is 
6o> 


(2) 


and  it  therefore  appears  that  the  mass  of  the  moving  charged  body 

2e2 
is  m'  +  ^r-  instead  of  only  m',  the  mass  of  the  uncharged  body. 

tjd 

The  second  term  represents  the  electromagnetic  mass.  The 
quantity  a  is  what  we  may  term  the  radius  of  the  electron.  This, 
however,  does  not  necessarily  mean  that  the  electron  is  a  well- 
defined  sphere  of  radius  a;  all  it  means  is  that  where  an  electron 
manifests  itself  the  modification  of  the  ether  is  such  as  would 
exist  if  a  charge  e  were  uniformly  distributed  over  the  surface 
of  a  sphere  of  radius  equal  to  a.  The  quantity  a  merely  represents 
one  of  the  limits  of  integration  arbitrarily  assumed  in  summing 
the  total  magnetic  energy  in  the  whole  space  through  which  the 
electron  moves. 

In  deriving  the  above  expression  for  the  energy  of  the  moving 
electron,  it  was  assumed  that  the  field  of  the  moving  electron 
is  the  same  as  that  of  the  stationary  electron.  This  is,  however, 
only  the  case  if  the  electron  moves  slowly,  because  when  a  Faraday 
tube  is  moved  it  tends  to  set  itself  at  right  angles  to  the  direction 
of  motion.  The  tubes  constituting  the  electron  therefore  tend  to 
crowd  together  in  a  plane  perpendicular  to  the  direction  of  motion 
of  the  electron.  The  result  is  an  increase  in  the  inertia  or  mass 
of  the  electron,  because  more  work  must  be  done  to  move  a 
Faraday  tube  parallel  to  itself  than  along  its  own  direction,  just- 
as  it  is  harder  to  move  a  log  of  wood  in  the  water  parallel  to 
itself  than  to  move  it  endwise.  This  increase  in  the  mass  of  the 
electron  only  becomes  appreciable  when  it  moves  with  a  speed 
greater  than  about  one- tenth  that  of  light;  for  speeds  less  than 
this  the  expression  (2) -can  be  taken  to  give  the  mass  of  the  elec- 
tron to  a  first  approximation,  but  for  higher  speeds  the  deter- 
mination of  the  mass  becomes  more  complicated.  The  mass  of 


8  THERMIONIC  VACUUM  TUBE 

the  electron  is  measured  by  the  ratio  of  the  force  to  the  accelera- 
tion to  which  it  gives  rise.  According  to  the  theory  of  Abraham 
and  Lorentz  the  electron  has  two  masses:  the  longitudinal  mass, 
when  it  is  accelerated  in  the  direction  of  motion,  and  the  trans- 
verse mass,  when  it  is  accelerated  perpendicular  to  the  direction 
of  motion  of  the  electron.  If  m  represents  the  mass  of  the  slow- 
moving  electron,  then  the  longitudinal  and  transverse  masses 
mi  and  m,2  are  given  by 

m\  _        1  7^2  _        1 

w 


where  c  is  the  speed  of  light.  It  is  seen  that  as  the  speed  of  the 
electron  approaches  that  of  light,  its  electromagnetic  mass  tends 
to  become  infinitely  large.  The  transverse  mass  of  the  high- 
speed electron  for  various  speeds  has  been  determined  by  Kauf- 
mann  and  Bucherer.1  Their  experiments  verify  the  above  ex- 
pression for  the  transverse  mass.  From  this  it  would  seem  that 
the  mass  of  the  electron  is  entirely  electromagnetic.  Later 
developments  of  the  Theory  of  Relativity  have  rendered  this  con- 
clusion somewhat  questionable,  so  that  there  does  not  seem  to  be 
definite  experimental  evidence  to  indicate  that  the  electronic 
mass  is  entirely  electromagnetic.2 

On  the  assumption  that  the  mass  is  entirely  electromagnetic 
equation  (2)  would  give  the  following  expression  for  the  simple 
mass  of  the  slow-moving  electron. 


(3) 


If  the  known  values  of  e  and  m  are  inserted  in  this  expression 
we  find  a  value  for  a  which  is  2X  10~13  cm.  This  effective  radius 
of  the  slow-moving  electron  is  therefore  only  about  one  fifty- 
thousandth  of  the  radius  of  the  hydrogen  atom. 

1W.  KAUFMANN,  Goott.  Nachr.  Math.-Phys.  Kl.,  p.  143,  1901;  p.  291, 
1902,  p.  90,  1903.  For  later  experiments  of  KAUFMANN,  Ann.  d.  Phys., 
Vol.  19,  p.  487,  1906.  A.  H.  BUCHERER,  Phys.  Zeitschr.,  Vol.  9,  p.  755,  1908. 

2  For  a  discussion  of  this  and  allied  questions,  the  reader  might  refer  to 
H.  A.  LORENTZ,  "  The  Theory  of  Electrons  "  and  L.  SILBERSTEIN,  "  The 
Theory  of  Relativity." 


PROPERTIES  OF  ELECTRONS 


9 


6.  Effect  of  Electric  Field  on  the  Motion  of  an  Electron. 

To  find  the  effect  of  an  electric  field  on  an  electron  is  a  compara- 
tively simple  matter  as  long  as  the  field  is  uniform.     Suppose  we 
have  two  infinitely  large  parallel  plates  OY  and  QR  (Fig.  3)  with 
a  potential  difference  V  between  them, 
and  let  the  electron  be  projected  with 
a  velocity  VQ  from  the  point  0  in  the 
direction    OQ.      On    account    of    the 
electric  field  between   the    plates  the 
velocity  of  the  electron  will  be  con- 
tinually increased    on    its  way  to  Q. 
The  kinetic  energy  of  the  electron  at 
the  moment  of  its  leaving  0  is  \mv<?\ 
if  its  velocity  on   reaching   Q  is  v  its' 
kinetic  energy  at  Q  will  be  %mv2.     The 
difference  in  these  values  of  the  kinetic 

energy  is  due  to  the  applied  electric  field.  If  E  is  the  intensity 
of  the  field  at  a  point  between  the  plates,  the  force  acting  on  the 
electron  is  Ee  and  the  work  done  on  it  is  ej  Eds,  where  ds  is 
an  element  of  the  path  along  OQ.  But  Eds  =  dV,  the  potential 
difference  between  the  ends  of  the  element  of  path  ds. 
Hence 


FIG.  3. 


ej  Eds  =  Ve  = 


(4) 


Now,  suppose  the  electron  be  projected  with  a  velocity  VQ 
in  the  direction  OP,  where  OP  makes  an  angle  0  with  the  direction 
OQ  of  the  field.  Since  the  only  force  acting  is  the  electric  force 
Xe,  we  have  for  the  equations  of  motion : 


Y  =  0 


d2x 


(5) 


where  m  is  the  mass  of  the  electron,  e  its  charge  and  X  the  intensity 
of  the  field  which  is  supposed  to  be  constant  and  uniform  between 
the  plates.  The  Y  component  of  the  initial  velocity  is  VQ  sin  <£, 
so  that 


y  =  vot  sn 


(6) 


10  THERMIONIC  VACUUM  TUBE 

Integrating  (5)  and  inserting  the  value  of  t  from  (6)  we  get: 

x  =  ^  -^2~+y  cot  4>  ......     (7) 

2m  vo2  sin2  <t> 

This  equation  gives  the  point  R  at  which  the  electron  will 
strike  the  plate  QR  when  the  distance  x  between  the  plates  and  the 
intensity  of  the  field  are  known.  If  the  electron  starts  from  0 
in  the  ctirection  OY,  <£  is  90°  and  equation  (7)  becomes 


This  equation  enables  us  to  calculate  the  deviation  x  of  an  electron 
from  its  path  by  an  electric  field  perpendicular  to  the  original 
direction  of  motion  of  the  electron. 

So  far  we  have  assumed  that  the  lines  of  force  between  the 
plates  are  straight.  If  this  is  not  the  case  the  motion  of  the 
electron  is  not  easily  determined.  The  case  in  which  the  electron 
moves  from  a  straight  wire  to  a  plate  is  one  in  which  the  field  is 
not  uniform.  Such  cases  are  frequently  met  with  in  the  study 
of  discharge  through  vacuum  tubes,  and  the  problems  involved 
become  so  difficult  that  the  desired  result  is  often  more  easily 
determined  empirically.  Such  is,  for  example  the  case  with  the 
three-electrode  thermionic  amplifier.  The  classification  of  cases 
dealing  with  electric  fields  that  can  be  represented  by  straight 
lines  of  force  and  which  can  be  handled  mathematically  is  a  purely 
geometrical  matter.  Such  fields  are  obtained  with  the  following 
structures:  (a)  Both  electrodes  are  infinitely  large  parallel 
plates;  (b)  one  electrode  is  an  infinitely  long  cylinder  and  the 
other  an  infinitely  long  wire  in  the  axis  of  the  cylinder;  (c)  both 
electrodes  are  infinitely  long  co-axial  cylinders;  (d)  one  electrode 
is  a  sphere  and  the  other  a  Doint  in  the  center  of  the  sphere;  (e) 
both  electrodes  are  concentric  spheres.  It  will  be  recognized  that 
in  all  these  cases  the  lines  of  electric  force  are  straight. 

7.  Effect  of  Magnetic  Field  on  the  Motion  of  an  Electron. 
Now,  instead  of  an  electric  field  let  us  apply  a  magnetic  field 
to  the  moving  electron.  As  was  shown  above  an  electron  moving 
with  a  velocity  v  is  equivalent  to  an  electric  current  i  —  ve.  We 
can  therefore  directly  apply  the  well-known  law  connecting  the 
mechanical  force  F  exerted  by  a  magnetic  field  of  intensity  H  on 
a  current  i\  namely,  F  =  nHi,  where  p  is  the  permeability  of  the 


PROPERTIES  OF  ELECTRONS  11 

medium.  Since  we  are  considering  the  motion  of  an  electron 
through  space  we  can  put'  /*  =  1,  so  that  the  force  on  the  electron  is 

F  =  Hev  .........     (9) 

This  force  is  at  every  instant  at  right  angles  to  both  the  direction 
of  motion  of  the  electron  and  that  of  the  magnetic  field.  Thus, 
referring  to  Fig.  3,  if  the  electron  starts  in  the  direction  OX  and 
the  magnetic  field  be  perpendicular  to  the  plane  of  the  paper 
and  directed  downwards,  the  electron  will  be  deviated  from  OX  in 
the  direction  OYr.  Now,  when  the  force  acting  on  a  body  is 
always  at  right  angles  to  its  direction  of  motion  the  body  must 
describe  a  circular  path,  the  force  being  given  by 


do) 


where  r  is  the  radius  of  curvature.     Hence,  we  get  from  (9)  and 
(10): 

mv 


This  equation  shows  how  strong  the  magnetic  field  must  be  to 
make  the  electron  travel  in  a  circle  of  any  desired  radius. 

Equation  (11)  expresses  an  interesting  and  useful  result. 
We  shall  mention  a  few  of  its  applications  here.  We  saw  above 
that  a  moving  electron  creates  a  magnetic  field  whose  lines  of  force 
are  circles  having  their  centers  along  the  path  of  the  electron. 
Now  consider  two  electrons  moving  side  by  side  in  the  same  direc- 
tion. Obviously  the  magnetic  field  produced  by  each  must 
exert  a  mechanical  force  on  the  other  in  the  sense  explained  above. 
A  consideration  of  the  directions  of  these  mechanical  forces  will 
show  that  the  two  moving  electrons  tend  to  attract  each  other. 
This  result  is  not  contrary  to  the  fundamental  law  of  electrostatics 
that  like  charges  are  repellent.  The  mutual  attraction  exerted  by 
the  electrons  is  due  only  to  their  motion  and  increases  with  their 
velocity.  If  they  moved  in  opposite  directions  they  would  repel 
each  other.  It  follows  from  this  that  an  electron  stream  in  a 
vacuum  tube  would  tend  to  shrink  together  if  the  velocity  with 
which,  the  electrons  move  become  sufficiently  high,  the  shrinkage 
increasing  with  the  velocity  with  which  the  electrons  comprising 
the  stream  move.  In  ordinary  cases  the  velocity  of  the  electrons 


12 


THERMIONIC  VACUUM  TUBE 


in  a  vacuum  tube  is  so  small  (of  the  order  of  a  million  centimeters 
per  second)  that  the  shrinkage  of  the  electron  stream  due  to  the 
reduction  in  the  mutual  electrostatic  repulsion  is  inappreciable. 

If  the  electron  source  in  a  vacuum  tube  is  a  hot  cathode  the 
electrons  are  emitted  from  it  in  all  directions :  the  electron  stream 
will  therefore  generally  spread  out  as  the  distance  from  the 
cathode  increases.  This  spreading  can  be  prevented  by  means  of  a 
magnetic  field  applied  in  a  suitable  way  to  the  stream.  In  Fig. 
4  let  C  be  the  hot  cathode,  P  an  anode  and  A  an  electrode  with 
an  aperture  in  its  center.  Let  A  and  P  be  connected  and  a  poten- 
tial difference  applied  between  them  and  the  cathode.  '  Of  the 
electrons  moving  away  from  the  cathode  some  go  to  A  and  some 
shoot  through  the  aperture  in  A  and  pass  on  to  P,  Between  C 


v 


Hi 


V-V 


FIG.  4. 


FIG.  5. 


and  A  their  velocity  will  be  continually  increased  by  the  electric 
field  existing  between  C  and  A,  but  after  passing  A  they  will 
continue  to  move  with  the  same  velocity  which  they  had  on 
reaching  A,  since  A  and  P  are  at  the  same  potential.  If  now  a 
magnetic  field  in  the  direction  AP  be  applied  by  means  of  a  coil 
as  shown  in  the  figure,  it  will  be  seen,  by  applying  the  above  laws, 
that  the  electrons  will  travel  along  a  helical  path,  the  diameter 
of  which  decreases  as  the  strength  of  the  magnetic  field  is  in- 
creased. In  Fig.  5,  H  represents  the  direction  of  the  magnetic 
field,  F  the  direction  of  the  force  on  the  electron,  and  S  the  path 
of  the  electron,  which  is  at  right  angles  to  F  and  H.  The  motion 
due  to  the  force  F}  when  added  to  the  primary  motion  in  the 
direction  of  H,  which  the  electron  has  when  passing  through  the 
aperture  in  A  of  Fig.  4,  results  in  the  electron  describing  a  helical 
path.  If  the  magnetic  field  be  made  sufficiently  strong  the 


PROPERTIES  OF  ELECTRONS  13 

diameter  of  the  helix  can  be  made  so  narrow  that  the  electrons 
practically  travel  in  a  straight  line  along  the  axis  of  the  tube. 

The  study  of  the  motion  of  an  electron  in  a  magnetic  field 
has  been  successfully  applied  to  the  determination  of  the  mass  of 
the  electron.  Referring  to  equation  (8)  it  is  seen  that  if  we  know 
the  velocity  VQ  with  .which  an  electron  moves  and  determine 
experimentally  the  extent  y  to  which  the  electron  stream  is  de- 
flected by  an  electric  field  X  whose  direction  is  at  right  angles 
to  the  direction  of  motion  of  the  electron,  we  can  calculate  the 
Q 

value  of  —  .     In  order  to  obtain  the  velocity  all  that  is  necessary 

wi 

is  to  apply  a  magnetic  field  in  such  a  way  as  to  counterbalance  the 
deflection  of  the  electron  stream  caused  by  the  electric  field.  Then 
the  magnetic  force  given  by  equation  (9)  must  be  equal  to  the 
electric  force  eX,  Hence 


(12) 


Millikan  has  accurately  determined  the  value  e  of  the  electronic- 

p 

charge.1     Hence,  knowing  e  and  —  we  can  obtain  the  mass  m  01 

the  electron.     This  value  has  been  found  to  be  9.01X  10~28  grm. 

8.  The  Accelerated  Electron.  Radiation.  We  have  seen  that 
an  electron  possesses  inertia.  'From  this  it  follows  that  in  order 
to  accelerate  an  electron  work  must  be  done  on  it  and  if  it  is 
retarded  in  its  motion  it  must  give  up  part  of  its  kinetic  energy. 
If  the  inertia  of  an  electron  is  wholly  electromagnetic  the  work 
done  in  accelerating  it  is  work  done  on  lines  of  force.  Suppose  a 
charge  with  its  connecting  lines  of  force  moves  through  space 
with  a  uniform  velocity.  If  this  charge  is  suddenly  retarded  the 
ends  of  the  lines  of  force  terminating  on  it  will  be,  so  to  speak, 
jerked  backwards.  In  accordance  with  the  properties  of  lines 
of  force  this  kink  created  at  the  end  of  the  line  will  not  be  trans- 
mitted along  the  whole  line  instantaneously  but  will  be  propagated 
along  it  with  a  finite  velocity  —  the  velocity  of  light.  These  kinks 
in  the  lines  are  the  seat  of  that  part  of  the  energy  which  the 
electron  gives  up  when  retarded.  It  can  be  shown  that  the 
electric  and  magnetic  forces  associated  with  a  .kinked  line  are 
more  intense  than  those  associated  with  a  straight  line.  In 

1  R.  A.  MILLIKAN,  "  The  Electron,"  University  of  Chicago  Press,  1917. 


14  THERMIONIC  VACUUM  TUBE 

the  latter  case  the  electric  force  at  a  distance  r  from  the  center 

£>  P(* 

of  «the  electron  is  -5  electrostatic  units,  or  -^  electromagnetic 

units,  c  being  the  velocity  of  light,  and  the  magnetic  force  that 
given  by  equation  (la).  If,  however,  an  electron  be  retarded  the 
electric  and  magnetic  forces  E  and  H  at  a  point  distant  r  from 
the  center  of  the  electron  at  the  moment  the  kink  passes  through 
that  point  are : 


(13) 


where  /  is  the  acceleration  and  6  the  angle  between  the  r  and 
the  direction  of  motion  of  the  electron.  E  and  H  are  at  right 
angles  to  each  other  and  to  the  direction  of  propagation  of  the 
kink  in  the  line.  The  energy  radiated  by  the  electron  is  there- 
fore radiated  as  electromagnetic  energy  with  the  speed  of  light. 

2e2/2 

The  total  amount  of  energy  radiated  by  the  electron  is  ~  — ^-. 

o   c 

If  now  such  an  electromagnetic  disturbance  passes  over  an 
electron  moving  with  uniform  velocity  the  electric  and  magnetic 
fields  associated  with  it  will  be  modified  by  the  intense  fields  in 
the  disturbances  and  this  modification  is  propagated  to  the  center 
of  the  moving  electron  along  the  lines  of  force  constituting  it. 
The  result  is  a  change  in  the  motion  of  the  electron.  It  is  seen 
therefore  that  the  energy  of  a  moving  electron  can  be  transformed 
in  co  radiation  energy  and  vice  versa,  the  transformation  always 
taking  place  when  the  electron  is  retarded  or  accelerated.  This 
result  is  an  important  agency  in  the  production  of  dislodged  elec- 
trons, that  is,  electrons  in  such  a  state  that  they  can  be  readily 
utilized  for  purposes  of  discharge  in  vacuum  tubes.  An  electron 
which  is  bound  to  an  atom  of  a  gas  or  vapor,  or  to  a  substance, 
can  be  dislodged  by  passing  an  electromagnetic  disturbance  in 
the  form  of  light  or  X-rays  over  it,  in  which  case  the  energy 
imparted  to  the  electron  may  be  so  great  that  it  can  overcome 
the  forces  that  bind  it  to  the  atom  or  substance. 

A  bound  electron  can  also  be  dislodged  by  arresting  the 
motion  of  a  high-speed  electron  in  its  neighborhood.  In  this  case 
the  kinetic  energy  of  the  moving  electron  is  first  transformed  into 


PROPERTIES  OF  ELECTRONS'  15' 

energy  of  radiation,  part  of  which  is  in  turn  transferred  to  the 
bound  electron. 

9.  Relation  between  Space  Charge  and  Potential  Distribution. 
In  dealing  with  the  conduction  of  electricity  by  dislodged  electrons 
or  positive  ions,  it  is  necessary  to  consider  the  effect  exerted  by 
their  presence  on  the  potential  distribution  between  the  electrodes. 
The  difference  between  the  number  of  electrons  and  positive  ions 
in  unit  volume,  multiplied  by  the  charge  per  ion,  is  usually  referred 
to  as  the  space  charge  or  volume  density  of  electrification. 

If,  in  the  space  between  two  electrodes,  there  are  no  positive 
ions,  and  n  electrons,  and  if  the  charge  on  the  positive  ion  and  the 
electron  be  eo  and  e,  then  the  distribution  of  potential  between 
the  electrodes  can  be  expressed  by 

Mnc-^o),   ....     (14) 

where  V  is  the  potential  at  a  point  having  the  coordinates  x,  y 
and  z.  This  equation  is  known  as  Poisson's  equation.  It  has  been 
used  extensively  in  investigations  dealing  with  the  conduction  of 
electricity  through  gases  and  high  vacua. 

In  applying  this  equation  to  the  case  in  which  the  charges 
are  contained  between  two  infinitely  large  parallel  plates,  between 
which  a  potential  difference  is  applied,  the  lines  of  force  are 
straight  and  everywhere  perpendicular  to  the  plates,  so  that  the 
equipotential  surfaces  are  planes  parallel  to  the  plates.  The  last 
two  terms  on  the  left-hand  side  of  equation  (14)  therefore  vanish 
and  we  get 

d2V 

.....     (15) 


where  p  is  the  volume  density  of  electrification  or  space  charge. 

If  there  are  no  free  charges  between  the  plates,  or  if  the  total 

positive  'charge  in  every  volume  element  is  equal  to  the  total 

dV 
negative  charge,  we   have  ne  —  nQUe  =  Q,  and  -7—  =  constant.    We 

then  have  the  simple  case  if  infinitely  large  parallel  plates  at  dif- 
ferent potentials,  but  with  no  charges  between  them,  in  which 
the  potential  at  different  points  is  a  linear  function  of  the  distance 
x  from  one  plate.  For  the  case  of  high  vacuum  tubes  in  which  the 
current  is  carried  almost  exclusively  by  electrons,  no  =  0  and 
=  ne. 


CHAPTER  II 

DISLODGMENT  OF  ELECTRONS  FROM  ATOMS  OF 
VAPORS  AND  GASES.     IONIZATION 

10.  Occurrence    of    Electrons.     In    this    and    the    following 
chapter    will    be    discussed   the   conditions   in    which   electrons 
normally  exist  and  the  means  whereby  they  can  be  brought  into 
such  a  state  that  they  are  readily  available  for   discharge  in 
vacuum  tubes. 

Since  all  charged  bodies  attract,  or  are  attracted  by,  oppositely 
charged  or  uncharged  bodies,  it  is  to  be  expected  that  there  are 
comparatively  few  electrons  floating  around  free  in  nature. 
By  far  the  larger  number  of  electrons  exist  as  the  building  stones 
of  which  all  matter  is  built  up,  and  they  are  held  in  this  condition 
by  very  strong  forces.  These  forces  are  due  to  the  positive  elec- 
trons in  the  nuclei  of  the  atoms.  Such  atomic  systems,  con- 
sisting of  electrons  and  positive  electrons,  are  electrically  neutral 
and  are  not  affected  by  an  electric  field.  The  fact  that  the 
conductivity  of  a  gas  or  vapor  is  very  small  is  an  indication  that 
there  can  only  be  very  few  electrons  in  the  gas  or  vapor  that  are 
not  bound  in  electrically  neutral  systems.  In  the  case  of  con- 
ducting solids  the  number  of  electrons  that  are  free,  or  that  can 
readily  be  made  free  by  the  application  of  an  electric  field,  is  com- 
paratively large,  so  that  such  solids  are  said  to  be  good  conduc- 
tors of  electricitjr.  But  such  electrons  cannot  be  said  to  be  dis- 
lodged. They  are  only  available  for  discharge  through  conductors 
and  not  for  discharge  between  conductors  separated  by  a'  gaseous 
or  vacuous  medium.  For  the  latter  purpose  they  must  be  dis- 
lodged not  only  from  the  atoms  in  the  substance  but  also  from  the 
substance  itself. 

11.  lonization.    The  process  of  the  production  of  dislodged 
electrons  is  known  as  ionization.     In  ah1  cases  this  process  involves 
overcoming  the  forces  that  hold  the  electrons  in  the  atoms  or  in 
the  substance.     In  accordance  with  the  properties  of  electrons 

16 


ELECTRONS  FROM  VAPORS  AND  GASES  17 

described  in  the  preceding  chapter  it  will  be  evident  that  this  can 
be  done  in  three  ways,  viz. :  (a)  by  means  of  the  impact  of  electrons 
or  positive  ions  on  the  atoms  or  substance;  (6)  by  means  of  electro- 
magnetic radiation:  (c)  by  means  of  heat.  The  first  of  these 
three  processes  gives  rise  to  the  phenomenon  of  delta  rays,  or 
secondary  electron  emission,  the  second  gives  rise  to  the  so-called 
photoelectric  effect,  and  the  third  forms  the  basis  of  the  subject 
of  thermionics.  The  present  chapter  will  be  devoted  to  a  dis- 
cussion of  the  phenomena  accompanying  the  dislodgment  of 
electrons  from  the  atoms  of  vapors  and  gases,  the  problems  of  the 
dislodgment  of  electrons  from  solid  substances  being  reserved 
for  the  next  chapter. 

12.  Constitution  of  the  Atom.  Although  very  little  is  known 
about  the  exact  nature  of  the  processes  going  on  in  an  atom  when 
its  equilibrium  is  disturbed,  there  are  nevertheless  a  certain  num- 
ber of  experimentally  determined  facts  giving  rise  to  theories 
that  successfully  account  for  many  of  the  phenomena  encountered 
in  the  ionization  of  atoms. 

There  is  very  little  doubt  but  that  the  atom  consists  of  a  number 
of  electrons  grouped  around  a  number  of  positive  electrons. 
The  absolute  value  of  the  positive  electronic  charge  is  the  same 
as  the  electron,  but  its  mass  is  1845  times  as  great.  The  positive 
electrons  in  an  atom  form  the  atomic  nucleus,  while  the  electrons 
are  separated  from  the  nucleus  by  distances  that  are  large  compared 
with  their  size.  For  our  present  purposes  it  does  not  matter  how 
these  mutually  repellent  positive  electrons  are  held  together  in 
the  nucleus;  it  is  likely  that  they  are  held  together  by  electrons 
so  that  the  nucleus  really  consists  of  a  group  of  positive  and  nega- 
tive electrons  having  a  resultant  positive  charge  equal  to  the  sum 
of  the  electronic  charges  outside  the  nucleus.  There  is  reason  to 
believe  that  the  electrons  grouped  around  the  positive  nucelus 
revolve  in  nearly  circular  orbits  round  the  nucleus.  If  the  elec- 
trons did  not  revolve,  the  force  of  attraction  between  them  and  the 
nucleus  would  cause  them  to  drop  into  the  nucleus.  On  the 
other  hand,  it  follows  from  ordinary  mechanics  that  when  an  elec- 
tron revolves  round  a  positive  nucleus  ic  must  be  constantly 
accelerated  and  must  therefore  be  constantly  radiating  energy. 
In  such  case  a  simple  system  like  the  hydrogen  atom,  which  con- 
sists of  a  single  positive  and  one  negative  electron,  would  radiate 
all  of  its  energy  in  such  a  short  time  that  it  really  could  not  exist 


18  THERMIONIC  VACUUM  TUBE 

at  all.  An  attempt  to  overcome  this  difficulty  has  been  made  by 
Bohr  by  making  an  assumption  which  frankly  repudiates  New- 
tonian mechanics  for  atomic  systems.  Bohr  assumes  that  although 
the  electrons  revolve  round  the  positive  nucleus,  it  does  not 
radiate  any  of  its  energy  as  long  as  it  remains  at  the  same  distance 
from  the  nucleus,  energy  being  only  radiated  or  absorbed  when 
this  distance  is  decreased  or  increased,  and  this  happens  only 
when  the  distance  is  changed  by  definite  amounts  so  that  the 
energy  is  radiated  or  absorbed  in  definite  quanta.  Bohr's  atom 
has  been  successful  in  explaining  and  predicting  a  number  of 
phenomena,  but  although  there  is  an  element  of  truth  in  it,  it  is 
still  far  from  the  whole  truth.  We  shall  therefore  not  enter  into 
any  further  discussion  of  it.  Suffice  it  to  say  that  the  necessity 
for  introducing  such  assumptions  as  Bohr's,  and  the  assumption  of 
energy  radiation  by  definite  quanta,  which  was  originated  by 
Planck,  seems  to  indicate  that  in  dealing  with  atomic  systems 
we  can  apply  the  Newtonian  system  of  mechanics  only  when  the 
atoms  are  in  a  steady  but  not  in  a  varying  state.  Since  Newtonian 
mechanics  was  built  up  on  an  experimental  basis  of  large-scale 
phenomena,  one  would  not  necessarily  expect  it  to  give  an  explana- 
tion of  atomic  phenomena. 

But  apart  from  the  question  of  the  behavior  of  the  electrons 
in  the  atom,  recent  experiments  have  given  conclusive  evidence 
that  the  atom  consists  of  a  number  of  electrons  held  together  in 
some  configuration  by  a  heavy  positive  nucleus.  The  total 
charge  of  the  nucleus  is  equal  to  the  sum  of  the  charges  of  the 
electrons,  so  that  the  atom  is  electrically  neutral.  The  total 
positive  charge,  or  the  number  of  electrons,  determines  the  chemi- 
cal nature  of  the  atom.  Starting  with  the  lightest  known  element, 
hydrogen,  all  the  elements  with  a  few  slight  deviations  are  obtained, 
in  the  order  of  their  atomic  weights,  by  successively  adding  one 
electron  and  the  equivalent  positive  charge. 

The  process  of  ionization  consists  in  the  detachment  of  one  or 
more  electrons  from  the  atom,  thus  leaving  the  atom  positively 
charged.  An  atom  from  which  one  or  more  electrons  have  been 
removed  is  known  as  a  positive  ion.  If  the  atoms  of  a  gas  be 
ionized  and  a  potential  difference  be  applied  between  two  plates 
immersed  in  the  gas,  the  positive  ions  will  move,  under  the  in- 
fluence of  the  electric  field,  to  the  negative  plate  and  the  electrons 
to  the  positive  plate.  If  the  pressure  of  the  gas  is  not  too  low 


ELECTRONS  FROM  VAPORS  AND  GASES  19 

and  the  speed  of  the  ions  or  electrons  not  too  high,  the  electrons 
that  have  been  detached  from  the  atoms  will  attract  other  neutral 
atoms  and  thus  form  negative  ions,  and  these  will  move  to  the 
positive  plate.  A  negative  ion  is  therefore  an  atom  which  has 
more  electrons  than  are  necessary  to  balance  the  charge  due 
to  the  positive  nucleus. 

In  order  to  ionize  an  atom  the  forces  that  hold  the  electrons 
to  the  nucleus  must  be  overcome.  These  forces  depend  partly 
on  the  distance  between  the  electron  and  the  positive  nucleus. 
Considering  the  case  of  an  atomic  system,  consisting  of  a  single 
positive  and  one  negative  electron,  the  mass  of  the  former  being 
very  much  larger  than  that  of  the  latter,  the  electron  can  revolve 
round  the  positive  electron,  or  escape  from  it,  according  as  its 
kinetic  energy  is  smaller  or  greater  than  its  potential  energy,  and 
in  the  formation  of  the  system  a  certain  amount  of  work  is  done 
by  the  electrical  forces  until  this  equality  is  attained.  This  energy 
of  formation  therefore  gives  a  measure  of  the  work  which  must  be 
done  to  remove  the  electron  from  the  nucleus,  and  will  be  greater 
the  smaller  the  distance  between  the  electron  and  the  nucleus. 
Secondly,  the  work  necessary  to  remove  the  electron  depends  on  the 
number  of  electrons  grouped  round  the  nucleus,  and  on  the  con- 
figuration of  the  system.  It  can  be  seen  in  a  general  way  that  if 
there  are  a  number  of  electrons  grouped,  for  example,  in  a  ring 
round  the  nucleus,  the  repulsion  exerted  by  the  other  electrons 
would  make  it  easier  to  remove  an  electron  than  would  be  the  case 
of  a  system  consisting  of  only  one  electron  and  one  positive 
electron.  Now,  the  way  in  which  the  electrons  are  grouped 
depends  upon  the  number  of  electrons  in  the  atom.  If  they  are 
grouped  in  rings  the  heaviest  atom  that  can  have  all  its  electrons 
in  one  ring  only  is  that  which  contains  eight  electrons,  namely,  the 
oxygen  atom.  Heavier  atoms  would  then  have  their  electrons 
arranged  in  two  rings,  still  heavier  in  three  rings,  and  so  on.  It 
can  therefore  be  seen  in  a  general  way  that  it  would  require  a 
smaller  expenditure  of  energy  to  detach  an  electron  from  an 
oxygen  atom  than  from  an  atom  of  hydrogen. 

13.  Radiation  from  Atoms  caused  by  Bombardment  of  Elec- 
trons. Let  us  now  look  into  the  process  of  ionization  in  greater 
detail.  Suppose  we  have  a  tube  containing  mercury  vapor  and 
two  electrodes,  A  and  B,  one  of  which  is  a  source  of  electrons. 
Let  a  potential  difference  be  applied  between  the  electrodes  so  that 


20  THERMIONIC  VACUUM  TUBE 

the  electrons  are  driven  from  the  one  to  the  other,  say,  from  A  to  B. 
On  their  way  these  electrons  will  collide  with  some  of  the  atoms 
of  the  vapor,  the  velocity  with  which  they  collide  increasing 
on  their  way  in  virtue  of  the  electric  field  between  the  electrodes. 
If  the  electrons  start  with  zero  velocity  from  the  electrode  A, 
their  velocity  v  after  having  dropped  through  a  potential  difference 
V  is  given  by  Ve  =  %mv*,  where  e  is  the  electronic  charge.  Now, 
it  has  been  found1  that  as  long  as  the  electrons  strike  the  atoms 
of  the  vapor  with  a  velocity  which  is  less  than  that  corresponding 
to  a  drop  through  a  certain  definite  voltage,  which,  in  the  case  of 
mercury,  is  about  5  volts,  they  are  reflected  from  the  atoms  with- 
out any  loss  of  energy.  The  impact  is  therefore  elastic.  If, 
however,  the  electrons  strike  the  atoms  with  a  velocity  greater  than 
this  value,  they  lose  part  or  all  of  their  energy,  and  at  the  same 
time  the  atom  radiates  energy  in  the  form  of  monochromatic 
light.  The  frequency  v  of  the  light  radiated  is  given  by  the 
following  relation: 

Ve  =  hv      ......     .-    .     (1) 

where  V  is  the  voltage  through  which  the  electron  has  dropped, 
and  h  ;s  Planck's  constant  of  action.  The  product  hv  has  the 
dimenj.,^  o>f  energy.  The  above  equation  expresses  one  of  the 
most  important  relations  of  modern  physics.  It  was  not  derived 
from  the  impact  experiments  of  Franck  and  Hertz  just  men- 
tioned; these  experiments  give  only  one  of  the  experimental 
verifications  of  the  relation.  It  was  originally  deduced  by  Ein- 
stein on  the  basis  of  Planck's  quantum  theory  of  radiation.  Ein- 
stein's equation  will  be  more  fully  discussed  when  we  come  to  con- 
sider the  photo-electric  effect. 

The  emission  of  light  in  the  form  of  monochromatic  radiation 
is  due  to  the  electron  in  the  atom  not  acquiring  sufficient  energy 
from  the  colliding  electron  to  get  out  of  reach  of  the  forces  of 
attraction  of  the  nucleus,  and  it  consequently  drops  back  to  its 
original  position,  thus  giving  up  the  energy,  in  the  form  of  mono- 
chromatic radiation,  which  it  has  acquired  from  the  colliding 
electron.  The  frequency  of  the  light  emitted  is  characteristic 
of  the  atom  and  is  referred  to  as  characteristic  radiation.  All 
known  atoms  have  a  large  number  of  characteristic  frequencies. 

1  FRANCK  AND  HERTZ,  Verb.  d.  D.  Phys.  Ges.,  Vol.  16,  pp.  457  and  512 
1914. 


ELECTRONS  FROM  VAPORS  AND  GASES        21 

These  frequencies  form  the  line  spectra  observed  in  the  discharge 
through  gases  and  vapors. 

The  blue  glow  observed  in  vacuum  tubes  that  are  not  well 
evacuated  is  due  to  the  impact  of  electrons  on  the  molecules  of  the 
residual  gas  and  is  the  resultant  of  a  large  number  of  character- 
istic frequencies  emitted  when  the  electrons  in  the  atoms  that  are 
displaced  by  the  colliding  electrons  drop  back  to  their  original 
positions  of  equilibrium.  Whenever  the  blue  glow  appears  some  of 
the  electrons  in  the  atoms  are  displaced  beyond  the  forces  of  attrac- 
tion of  the  atoms  and  take  part  in  the  current  convection  through 
the  tube.  This  process  of  completely  detaching  electrons  from  the 
atoms  by  means  of  colliding  electrons  is  known  as  ionization  by 
collision.  The  blue  glow  in  vacuum  tubes  is  therefore  always 
an  indication  that  ionization  by  collision  takes  place. 

As  was  explained  in  the  first  chapter,  an  electron  can  also 
acquire  energy  from  a  light  wave  passing  over  it.  And  since 
the  energy  in  a  wave  and  the  energy  of  an  electron  are  related  as 
shown  by  equation  (1),  it  follows  that  a  wave  of  frequency  v  and 
produce  the  same  effects  explained  above  that  are  produced  by  an 
electron  that  has  dropped  through  a  voltage  F,  where  F  and  v 
are  related  by  equation  (1). 

14.  Ionization  Voltage  and  Convergence  Frequency.  The 
least  energy  with  which  an  electron  must  collide  with  an  atom 
in  order  to  completely  detach  an  electron  from  the  atom  of  any 
gas  or  vapor  is  known  as  the  ionization  energy  of  the  gas  or  vapor. 
This  amount  of  energy  is  usually  expressed  in  terms  of  the  voltage 
through  which,  the  electron  drops  before  it  collides  with  the  atom, 
and  is  then  referred  to  as  the  ionization  voltage.  The  ionization 
voltage  is  the  ionization  energy  divided  by  the  charge  of  the  elec- 
tron. 

When  an  electron  in  the  atom  is  displaced  to  such  an  extent 
that  the  force  of  attraction  of  the  parent  atom  just  manages 
to  pull  it  back  to  its  original  position  within  the  atom,  a  character- 
istic radiation  is  emitted  whose  frequency  is  known  as  the  converg- 
ence frequency.  It  is  the  shortest  wave  length  that  can  be  emitted 
by  the  most  loosely-bound  electrons  in  the  normal  atom.  Apply- 
ing equation  (1)  we  can,  if  we  know  the  convergence  frequency, 
compute  the  ionization  voltage.  This  relatign  is  important  be- 
cause it  is  often  easier  to  obtain  the  ionization  voltage  by  measur- 
ing the  convergence  frequency  from  observation  of  the  line  spectra 


22  THERMIONIC  VACUUM  TUBE 

of  the  gas  or  vapor  than  by  measuring  the  ionization  voltage 
itself.  Direct  determinations  have  been  made  of  the  ionization 
voltage  of  various  gases  and  vapors,  but  there  is  reason  to  believe 
that  most  of  the  values  obtained  are  not  reliable.  The  following 
values  of  the  ionization  voltage,  computed  from  the  convergence 
frequencies,  give  an  idea  of  the  order  of  magnitude  of  this  important 
quantity. 

Ionization  Voltage 
Substance.  Voltg 

Mercury  vapor 10 . 4 

Zinc  vapor 9 . 24 

Magnesium  vapor 9.13 

Calcium  vapor . 9 . 96 

Helium 29.00 

Hydrogen 13.6 

Since  an  electron  can  ionize  a  gas  atom  after  having  dropped 
through  the  ionization  voltage  of  the  gas,  it  follows  that  in  a  vac- 
uum tube  which  contains  some  residual  gas,  ionization  always 
takes  place  if  the  voltage  applied  between  cathode  and  anode 
exceeds  the  ionization  voltage.  Now,  it  is  impossible  completely 
to  remove  the  last  traces  of  gases  and  vapors  from  a  vacuum  tube. 
Hence,  in  thermionic  tubes  operating  on  applied  voltage  greater 
than  those  given  in  the  above  table  some  ionization  by  collision 
always  takes  place,  although  the  amount  of  ionization  in  well- 
evacuated  tubes  may  not  be  large  enough  to  cause  an  appreciable 
effect  on  the  operation  of  the  lube.  A  discharge  which  is  carried 
entirely  by  electrons  is  a  pure  electron  discharge.  When  the  term 
is  applied  to  the  discharge  through  vacuum  tubes,  as  actually 
realized  in  practice,  it  does  not  necessarily  mean  a  discharge  which 
is  carried  entirely  by  electrons,  but  one  in  which  the  number  of 
positive  ions  formed  by  collision  ionization  is  so  small  as  not  to 
have  any  appreciable  influence  on  the  operation  of  the  tube. 


CHAPTER  III 

DISLODGMENT  OF  ELECTRONS  FROM  SOLID 
SUBSTANCES 

15.  Free  Electrons.     If  a  substance  contains  electrons  that  are 
not  bound  to  atoms  to  form  electrically  neutral  systems,  the  sub- 
stance must  be  conducting,  because  if  it  were  placed  in  an  electric 
field  the  free  electrons  would  move  in  the  direction  of  the  field 
and  thus  establish  a  current  in  the  substance.     In  order  to  account 
for  the  conductivity  of  metallic  substances  the  assumption  has  been 
made  that  metals  contain  a  large  number  of  free  electrons.     This 
assumption  has  been  questioned.     On  the  other  hand,  the  fact 
that  a  substance  conducts  electricity  indicates  that  it  must  enable 
electrons  to  pass  freely  through  it  under  the  application  of  an 
electric  field.     It  is  possible  that  the  conductivity  of  metals  is 
due  to  the  frequency  of  collision  of  the  atoms  of  the  metal  with  each 
other.     When  two  atoms  collide  there  is  a  chance  that  an  electron 
originally  belonging  to  one  of  the  atoms  comes  so  well  within  the 
field  of  force  of  the  other  atom  that  it  is  attracted  with  equal 
forces  by  both  atoms,  so  that  the  resultant  force  on  the  electron 
is  very  small.     The  electron  would  therefore  be  essentially  free 
at  that  moment  and  if  the  metal  were  placed  in  an  electric  field 
by  applying  a  potential  difference  between  its  ends,  the  electron 
would  move  in  the  direction  of  the  field.     This  would,  of  course, 
leave  one  of  the  atoms  positively  charged,  but  its  loss  would 
immediately  be  compensated  for  by  electrons  coming  from  the 
source   of   potential   difference.     On   account   of   the   frequency 
of  collisions  of  the  atoms  in  the  metal  a  large  number  of  electrons 
can  thus  be  set  free  and  made  to  move  along  the  lines  of  force  of 
an  applied  field.     In  the  case  of  gases  where  the  atoms  are  rela- 
tively far  apart,  the  chance  of  this  happening  is  very  small,  so  that 
a  gas  is  not  a  good  conductor  of  electricity. 

16.  Force  that  Holds  Electrons  in  Substance.    The  electrons 
and  atoms  of  a  solid  substance,  like  the  atoms  of  a  gas,  possess 

23 


24  THERMIONIC  VACUUM  TUBE 

kinetic  energy  and  are  in  a  constant  state  of  motion.  Now, 
if  the  electrons  in  a  substance  possess  kinetic  energy  the  ques- 
tion arises  why  do  they  not  escape  from  the  substance.  The 
answer  to  this  question  is  an  assumption  that  there  exists 
at  the  surface  of  the  substance  a  force  which  tends  to  keep 
the  electrons  in  the  substance.  Being  an  assumption  it  is 
necessarily  a  very  unsatisfactory  answer.  But  there  is  a  good 
reason  to  believe  that  this  assumption,  which  was  made  by  O.  W. 
Richardson  in  1901,  is  one  which  did  not  lead  us  astray.  In  fact 
recent  developments  regarding  the  structure  of  the  atom  lead  us 
to  believe  that  such  a  force  which  tends  to  hold  electrons  in  a 
substance  must  necessarily  exist  at  the  surface  of  the  substance, 
and  Richardson's  assumption  can  be  explained  in  a  manner 
which  is  entirely  consistent  with  our  physical  conceptions.  In 
order  to  escape  from  the  surface  of  the  substance  an  electron 
must  do  work  in  overcoming  the  force  which  tends  to  hold  it  in  the 
substance,  and  this  amount  of  work  it  does  at  the  expense  of  its 
own  kinetic  energy.  For  all  known  substances  the  work  which 
an  electron  must  do  to  escape,  and  the  amount  of  kinetic  energy 
possessed  by  the  electrons  in  the  substances  are  of  such  order  of 
magnitude  that  only  very  few  electrons  manage  to  escape  at 
ordinary  temperatures.  By  far  the  larger  number  of  them  would 
have  to  expend  more  energy  than  they  possess,  so  that  they  are 
held  within  the  substance. 

The  work  which  an  electron  must  do  to  escape  from  the  surface 
of  a  substance  is  sometimes  referred  to  as  the  "  electron  evaporation 
constant."  Generally  it  is  expressed  in  terms  of  equivalent  volts. 
The  evaporation  constant  w  and  the  equivalent  voltage  </>  are  con- 
nected by  the  relation 


w 


where  e  is  the  electron  charge.  We  shall  in  the  following  refer  to  0 
as  the  electron  affinity.  This  quantity  is  the  most  important  con- 
stant in  thermionics.  It  determines  the  thermionic  current  that 
can  be  obtained  from  any  particular  type  of  cathode  at  any  desired 
temperature,  and  is  characteristic  of  the  substance  used  as  cathode. 
The  smaller  <£  is,  the  larger  is  the  thermionic  current  that  can  be 
obtained.  It  is  desirable  that  cathodes  be  used  in  thermionic 
tubes  for  which  <£  is  as  small  as  possible,  because  the  power  that 


ELECTRONS  FROM  SOLID  SUBSTANCES  25 

must  be  dissipated  in  the  heating  of  the  cathode  to  obtain  a  definite 
thermionic  current  decreases  as  <£  is  decreased.  This  implies 
economy  of  operation  as  well  as  increased  life  of  the  tube,  be- 
cause of  the  lower  temperature  at  which  the  cathode  can  be 
operated.  The  coated  type  of  cathode  (Wehnelt  cathode)  is  an 
example  of  a  cathode  that  has  been  so  treated  as  to  obtain  a  low- 
value  of  the  electron  affinity. 

To  obtain  a  better  insight  into  the  nature  of  the  electron 
affinity,  let  us  consider  an  evacuated  enclosure  divided  into  two 
parts  A  and  B  by  a  surface  S.  Let  A  and  B  each  contain  a  number 
of  electrons  arid  suppose  that  the  electrons  in  B  can  pass  freely 
through  the  surface  into  A ,  but  in  passing  from  A  to  B  they  must 
do  a  certain  amount  of  work  w.  The  electrons  in  A  and  B  possess 
kinetic  energy  like  the  molecules  of  a  gas;  they  will  therefore 
move  about  at  random,  continually  passing  through  the  surface  s 
in  either  direction.  The  steady  state  will  be  reached  when  as  many 
electrons  pass  per  second  from  A  to  B  as  from  B  to  A.  When  this 
state  is  attained  there  will  be  more  electrons  in  A  than  in  B. 
The  relation  between  these  numbers  is  given  by  Boltzman's 
equation: 

w  A7 

n  =  Ne~kT    or    w  =  kTlog— ,   .     .     .     .     .     (2) 

where  N  and  n  denote  the  number  of  electrons  per  unit  volume  in 
A  and  B  respectively,  T  the  temperature  cf  the  system,  w  the  work 
which  an  electron  must  do  to  move  from  A  to  B  and  k  the  gas  con- 
stant per  electron,  sometimes  referred  to  as  Boltzmann's  constant. 
This  constant  k  is  two-thirds  of  the  average  kinetic  energy  which 
an  electron  possesses  at  a  temperature  of  1°  absolute  and  is  equal 
tol.36Xl(T16erg. 

We  can  now  replace  the  part  A  within  our  enclosure  by  a  metal, 
so  that  the  surface  S  is  the  surface  of  the  metal.  The  replacement 
is  entirely  in  accordance  with  our  fundamental  assumption  that  in 
escaping  from  the  metal  an  electron  must  do  a  certain  definite 
amount  of  work.  The  number  of  electrons  immediately  outside 
the  surface  of  the  metal  will  then  be  related  to  the  number  inside 
the  metal  by  the  above  equation  (2).  It  is  seen  then  that  the 
number  n  that  escape  from  the  metal  depend  upon  the  two  factors 
T  and  w,  and  that  this  number  increases  as  the  temperature  of  the 
system  is  increased  or  as  w  is  decreased.  Moreover,  it  is  seen 


26  THERMIONIC  VACUUM  TUBE 

that,  since  both  T  and  w  appear  in  the  exponent  of  €,  a  small 
change  in  either  of  them  causes  a  considerable  change  in  the 
number  of  electrons  that  escape  from  the  metal.  If,  on  the  other 
hand,  both  T  and  w  be  kept  constant,  the  number  which  escapes 
increases  as  the  number  of  electrons  N  in  the  metal  is  increased. 
This  increase  is,  however,  not  nearly  as  effective  as  that  occasioned 
by  a  decrease  in  w.  These  considerations  show  the  importance 
of  the  constant  w.  Its  influence  on  the  phenomena  encountered 
in  the  thermionic  discharges  goes  even  further  than  this,  as  we 
shall  now  proceed  to  show. 

17.  Contact   Electromotive   Force.    The   advent  of  galvanic 
electricity  was  the  discovery  by  Volta,  over  a  hundred  years  ago, 
that  when  two  pieces  of  different  metals  were  placed  in  a  contact 
and  then  separated  they  acquire  electric  charges.     If  the  two 
pieces  of  different  metals  were  placed  in  an  electrolyte  and  joined 
by  a  wire  outside  the  electrolyte  a  current  is  established  in  the 
circuit  so  formed.     The  nature  of  this  force  which  drives  elec- 
tricity round  the  circuit  and  is  known  as  contact  electromotive 
force  was  never  understood  until  recently.     It  was  only  in  the  last 
two  decades  that  it  was  shown  conclusively,  mainly  by  O.  W. 
Richardson  and  P.  Debije,  that  the  contact  electromotive  force 
is  an  intrinsic  property  of  metals  and  is  determined  by  the  electron 
evaporation  constant  w.     The  connection  between  the  contact 
E.M.F.  and  the  evaporation  constant  w  can  be  gathered  from  the 
following.     Since  an  electron  must  do  work  in  escaping  through 
the  surface  of  a  metal,  it  follows  that  two  points,  one  outside  of 
the  surface  and  the  other  inside,  must  be  at  different  potentials. 
This  difference  of  potential  is  given  by  equation  (1).     Now  suppose 
we  have  two  slabs  of  different  material,  such  as  copper  and  zinc. 
Let  them  be  connected  by  a  copper  wire  as  shown  in  Fig.  6,  and  let 
the  number  of  electrons  per  unit  volume  in  the  copper  and  zinc 
be,  respectively,  NI  and  AT2.     Since  the  two  pieces  of  metal  are 
metallically  connected  all  points  in  the  metallic  circuit  must  be 
at  the  same  potential,  except  for  a  very  small  potential  difference 
which  occurs  at  the  junction  AB  of  the  two  metals.     Let  the 
circuit  be  grounded,  so  that  it  can  be  considered  to  be  at  zero 
potential.     Let  the  work  which  an  electron  must  do  to  escape  from 
the  copper  slab  be  w\ ,  so  that  in  moving  from  the  metal  to  a  point 
just  outside  the  surface  its  potential  changes  from  zero  to  a  value 
Vi,  say.     Let  the  corresponding  values  for  the  zinc  slab  be  W2 


ELECTRONS  FROM  SOLID  SUBSTANCES 


27 


and  V2,  and  let  us  see  how  much  work  must  be  done  in  moving 
an  electron  from  a  point  in  the  copper  through  the  space  to  a  point 
inside  the  zinc.  In  moving  through  the  surface  of  the  copper  an 
amount  of  work,  wi,  must  be  done  by  the  electron.  In  moving 
from  here  to  a  point  just  outside  the  zinc  surface  it  does  an  amount 
of  work  equal  to  (Vi  —  V2)e,  where  e  is  the  charge  of  the  electron, 
and  in  moving  through  the  surface  of  the  zinc  slab  the  work  done 
is  —W2.  The  total  amount  of  work  done  is  therefore 


Cu 


FIG.  6. 


Now,  since  the  number  of  electrons  per  unit  volume  in  the  copper 
and  the  zinc  is  NI  and  N2,  respectively,  we  have  by  equation  (2) : 

kT         N-,     <j/)o  —  i/)t 

TT  TT  A/ J.        <•  IV    1         ,        W> '  A  ™J\  /O\ 

Vi  —  V2  = log-TT — •     '••••(«*) 

e      &  N2         e 

kT       TVi 
The  term  —  log  -^  gives  rise  to  the  small  E.M.F.  set  up  at  the 

junction  AB  of  the  metals  and  accounts  for  the  Peltier  effect. 
It  is  so  small  in  comparison  with  the  other  terms  in  equation  (3) 
that  it  can  be  neglected.  Hence,  referring  to  equation  (1),  it 
follows  that 

The  difference  Vi  —  V2  is  called  the  contact  potential  difference 
between  the  two  metals,  and,  as  is  seen  from  equation  (4),  it  is 
equal  to  the  difference  between  the  electron  affinities  of  the  metals. 


28  THERMIONIC  VACUUM  TUBE 

18.  Measurement  of  Contact  E.M.F.  That  this  difference  of 
potential  actually  exists  between  the  two  metals  can  be  shown  in 
the  following  way:  Suppose  the  circuit  be  cut  at  a  place  C,  and 
the  two  ends  connected'  to  the  quadrants  of  a  quadrant  electrom- 
eter. Let  us  connect  one  of  the  plates  (say,  the  copper  plate) 
and  the  corresponding  pair  of  quadrants  to  ground,  while  the  other 
side  of  the  system  remains  insulated.  The  system  constituted  by 
the  plates  Cu  and  Zn,  and  the  quadrants,  has  a  definite  electro- 
static capacity  depending  on  the  distance  between  the  plates. 
If  the  plates  first  be  placed  close  together  and  then  jerked  apart, 
the  capacity  of  the  system  will  change,  and  if  a  potential  differ- 
ence exists  between  the  plates  the  electrometer  will  show  a  deflec- 
tion. If  the  potential  difference  is  P  and  the  deflection  di,  the 
sensitivity  of  the  measuring  system  is  given  by 


Now,  instead  of  directly  grounding  the  copper  plate,  let  it 
be  connected  to  ground  through  a  battery  which  maintains  it  at 
a  constant  potential  V,  so  that  the  potential  difference  between  the 
two  plates  is  P+  V.  If  the  plates  are  now  placed  the  same  distance 
apart  as  they  were  initially  in  the  first  operation,  and  then  again 
pulled  apart  to  the  same  extent  as  before,  the  electrometer  will 
show  a  different  deflection  cfe,  and  the  sensitivity  of  the  measuring 
system  is  now  given  by 


p+r 

Equating  these  two  expressions  for  the  sensitivity,  the  contact 
potential  difference  P  between  the  two  plates  is : 

P=v  * 


We  shall  see  later  that  the  contact  potential  difference,  and 
hence  also  the  electron  affinity,  depends  very  much  on  the  nature 
of  the  surface  of  the  substance.  It  can  be  modified  very  appre- 
ciably by  gas  occluded  in  the  surface.  The  notable  effect  of  gas 
has  led  some  to  believe  that  the  contact  potential  difference  is  not 
an  intrinsic  property  of  metals  but  is  occasioned  entirely  by  the 


ELECTRONS  FROM  SOLID  SUBSTANCES  29 

presence  of  gas.  There  is,  however,  no  doubt  but  that  contact 
potential  difference  must  exist  between  metals  in  the  best  obtain- 
able vacuum,  and  that  it  is  determined  by  the  electron  affinity. 
A  film  of  gas  on  the  surface  of  the  substance  can  increase  or 
decrease  the  electron  affinity  and  so  change  the  contact  potential 
difference  between  it  and  another  substance.  It  will  be  shown 
later  that  this  also  produces  a  change  in  the  thermionic  current 
obtainable  from  the  substance.  Thus,  when  hydrogen  is  occluded 
in  the  surface  of  platinum  the  work  which  an  electron  must  do  to 
escape  from  the  surface  of  the  platinum  is  decreased,  while 
oxygen  occluded  in  the  surface  of  calcium  increases  it. 

The  following  table  gives  the  electron  affinities  for  a  number 
of  substances,1  expressed  in  volts: 

Tungsten 4.52  Zinc 3.4 

Platinum 4.4  Thorium 3.4 

Tantalum 4.3  Aluminium 3.0 

Molybdenum 4.3  Magnesium 2.7 

Carbon 4.1  Titanium 2.4 

Silver 4.1  Lithium 2 . 35 

Copper 4.0  Sodium 1 .82 

Bismuth 3.7  Mercury 4.4 

Tin 3.8  Calcium 3.4 

Iron 3.7 

The  difference  between  any  two  of  these  values  gives  the  con- 
tact potential  difference  between  the  corresponding  substances. 

We  shall  see  that  the  contact  potential  difference  plays  an 
important  part  in  thermionic  amplifiers  and  detectors  of  electro- 
magnetic waves  that  are  so  designed  as  to  operate  on  small  plate 
voltages. 

The  values  of  electron  affinities  given  in  the  above  table  are 
of  such  order  of  magnitude  that  under  normal  conditions  only  very 
few  of  the  electrons  in  the  substance  possess  sufficient  kinetic 
energy  to  enable  them  to  escape  by  overcoming  the  force  of  attrac- 
tion at  the  surface.  In  order  to  make  use  of  electrons  for  the 
purpose  of  discharge  through  vacuum  tubes  they  must  first  be 
dislodged  from  the  parent  substance. 

We  shall  now  proceed  to  a  discussion  of  the  agencies  whereby 

1  Most  of  these  are  averaged  values  compiled  by  LANGMUIR  (Trans.  Am. 
Electro-chem.  Soc.,  Vol.  29,  p.  166,  1916)  from  measurements  of  RICHARD- 
SON, MILLIKAN,  HENNING,  LANGMUIR,  and  others. 


30 


THERMIONIC  VACUUM  TUBE 


the  dislodgment  of  the  electrons  can  be  effected.  As  was  stated 
in  Section  (11)  these  agencies  are:  (1)  heat;  (2)  electromagnetic 
radiation,  and  (3)  impact  of  electrons.  These  agencies  form  the 
basis  of  the  subjects  of  Thermionics,  Photo-electricity  and  Second- 
ary Electron  Emission,  respectively. 

19.  Elements  of  Thermionics.  The  first  of  these  agencies 
which  is  the  most  important  for  our  immediate  purposes  has  been 
known  for  a  considerable  time.  In  fact,  it  has  been  known  for  over 
one  hundred  years  that  when  a  metal  is  brought  into  a  state  of 
incandescence  the  air  in  its  neighborhood  becomes  a  conductor  of 
electricity.  The  phenomenon  was  studied  in  detail  by  Elster 
and  Geitel  during  the  years  of  1882-1889.  They  found  that 
when  a  metallic  filament  was  placed  near  a  plate  the  latter  acquired 
a  charge  when  the  filament  was  heated  to  incandescence.  At  red 
heat  the  plate  acquired  a  positive  charge,  but  when  the  tempera- 
ture of  the  filament  was  raised  to  white  heat  the  plate  charged  up 

negatively.  If  the  filament  and  plate 
were  placed  in  an  enclosure  which  could 
be  evacuated,  the  tendency  for  the  plate 
to  charge  up  negatively  was  increased. 

This  effect  also  came  to  the  notice 
of  Edison  in  1883.  He  noticed  that  if 
a  metallic  plate  be  inserted  in  the 
vacuous  space  of  an  incandescent  lamp 
and  this  conductor  be  connected  to  the 
positive  end  of  the  filament,  a  current 
was  established  in  the  shunt  circuit  so 
formed,  namely,  the  circuit  PFG  (Fig. 
7).  But  if  the  plate  was  connected  to 
the  negative  end  of  the  filament,  the 
galvanometer  showed  no  deflection.  A 
study  of  this  effect,  which  is  sometimes 
called  the  "  Edison  Effect  "  was  made 
by  J.  A.  Fleming  in  1896,1  but  the  true 
nature  of  the  phenomenon  was  not 

understood  until  the  work  of  J.  J.  Thomson  and  O.  W.  Richardson. 
In  1899,  the  former  showed  2  that  the  phenomenon  was  the  result 
of  negative  electricity  given  off  from  the  hot  filament  in  the  form 

1  J.  A.  FLEMING,  Phila.  Mag.;  Vol.  42,  p.  52,  1896. 

2  J.  J.  THOMSON,  Phil.  Mag.,  Vol.  48,  p.  547,  1899. 


FIG.  7. 


ELECTRONS  FROM  SOLID  SUBSTANCES  31 

of  electrons.  This  explained,  for  example,  why  in  the  Edison 
effect  a  current  was  observed  to  flow  through  the  galvanometer 
(7.  When  the  plate  was  connected  to  the  positive  end  of  the 
filament,  the  filament  had  a  negative  potential  with  respect  to  the 
plate,  and  the  electrons  given  off  by  the  filament  were  driven  to 
the  plate.  Since  the  time  of  Thomson's  experiment  there  has  been 
no  doubt  in  the  minds  of  physicists  that  the  carriers  of  electricity 
from  the  filament  are  electrons,  but  the  mechanism  of  the  emission 
of  these  electrons  from  the  hot  filament  was  not  known  until 
O.  W.  Richardson  l  showed,  in  1901,  that  the  electrons  are  emitted 
solely  in  virtue  of  their  kinetic  energy  and  need  no  chemical  reac- 
tion at  the  surface  of  the  filament.  This  result  of  Richardson's 
work  was  the  first  definite  expression  of  what  may  be  termed  a 
pure  electron  emission. 

Richardson's  theory  was  based  on  an  assumption  that  had 
previously  been  made  and  successfully  applied,  that  the  electrons 
in  a  metal,  which  are  free  to  move  under  the  influence  of  an 
electric  field,  behave  like  the  molecules  of  a  gas,  that  is,  they 
have  velocities  distributed  according  to  Maxwell's  law.  It  was 
stated  in  Section  16  that  these  electrons  are  held  in  the  substance 
by  a  force  existing  at  the  surface  of  the  substance.  There  is  still 
some  speculation  regarding  the  exact  nature  of  this  force  which 
seems  to  be  closely  related  to  the  structure  of  the  atoms  or  mole- 
cules of  the  substance.  At  ordinary  atmospheric  temperatures 
very  few  electrons  possess  sufficient  kinetic  energy  to  overcome 
this  force.  The  number  escaping  at  such  temperatures  is  there- 
fore extremely  small.  According  to  Maxwell's  law  of  velocity 
distribution  some  electrons  will  at  one  moment  have  zero  velocity, 
others  again  will  have  extremely  high  velocities,  while  the  majority 
will  possess  velocities  ranging  between  these  two  extreme  values. 
Only  the  few  electrons  with  the  very  high  velocities  will  be  able  to 
escape  through  the  surface.  The  energy  w  which  an  electron  must 
expend  to  overcome  the  force  of  attraction  at  the  surface  is  related 
to  the  number  of  electrons  per  cubic  centimeter  inside  and 
outside  the  surface  by  equation  (2).  From  this  equation  it  is  seen 
that  as  the  temperature  is  raised  the  number  n  of  electrons  outside 
the  surface  increases.  Now,  in  vacuum  tubes  we  are  not  so  much 
concerned  with  the  relative  number  of  electrons  inside  and  outside 

1 0.  W.  RICHARDSON,  Proc.  Camb.  Phil.  Soc.,  Vol.  11,  p.  286,  1901.     Phil. 
Trans.  Roy.  Soc.,  Vol.  11,  p.  497,  1903. 


32  THERMIONIC  VACUUM  TUBE 

of  the  surface  in  the  state  of  equilibrium  as  with  the  rate  at  which 
they  escape  when  they  are  carried  away  as  fast  as  they  are  emitted. 
This  can  be  ascertained  by  applying  a  potential  difference  between 
the  body  which  emits  the  electrons  and  a  conductor  placed  in  its 
neighborhood.  It  will  be  understood  in  the  following  that  this 
potential  difference  is  always  high  enough  to  draw  all  the  electrons 
away  as  fast  as  they  are  emitted.  Applying  the  principles  of  the 
kinetic  theory  of  gases  it  can  be  shown  that,  to  an  approximation 
which  is  sufficiently  close  for  our  purposes,  the  number  n'  of  elec- 
trons that  pass  per  unit  time  through  unit  area  of  the  surface  from 
the  inside  is  given  by: 


IkT 

n  —tt\/o — , 
\27rra' 


kT 

(5) 


where  m  is  the  mass  of  the  electron,  and  n  is  the  number  of  elec- 
trons per  cubic  centimeter  outside  the  surface. 

This  number  of  electrons  can  be  obtained  in  terms  of  the 
number  N  per  cubic  centimeter  inside  the  surface  by  combining 
the  relation  (5)  with  equation  (2).  Thus: 


IrT 

.         kT. 
27rra 


n'  is  the  number  of  electrons  that  would  move  per  second  to  a 
conductor  which  is  charged  positively  and  placed  in  the  neighbor- 
hood of  the  emitting  substance.  If  e  be  the  electronic  charge, 
then  n'e  is  the  saturation  current  per  square  centimeter  surface 
of  the  emitting  substance,  or 


or 

^ 
7,=A!T1/2€  T  .........    -     (7) 

The  constant  b  in  this  equation  is  a  temperature  and  is  ex- 
pressed in  absolute  (Kelvin)  degrees.  It  is,  however,  more  con- 
venient to  use  the  equivalent  constant  <f>  expressed  in  volts.  The 
relation  between  </>  and  6  is  as  follows  :  From  equations  (6)  and  (7) 
w  =  kb,  and  from  equation  (1)  w  =  <l>e,  where  <f>  is  expressed  in  elec- 
trostatic units,  or 

u 
volts  .........     (8) 


ELECTRONS  FROM  SOLID  SUBSTANCES 


33 


Now  k  is  the  gas  constant  for  one  electron  and  is  equal  to 
1.36X10"16,  while  e  is  the  electronic  charge  in  electrostatic 
units.  Hence 


. 
<f>  = 


u 
volts, 


i.e., 


=  8.6XlO-56volts. 


(9) 


The  constant  <£  is  the  electron  affinity,  values  of  which  for  a 
number  of  substances  are  given  in  the  table  on  p.  29.  It  can 
be  determined  experimentally  with  a  simple  device  consisting  of 
a  filament  of  the  substance  to  be  investigated  and  an  anode  placed 
in  its  neighborhood,  the  structure  being  enclosed  in  a  vessel 
which  can  be  evacuated  to  such  an  extent  that  the  residual  gas 


Temperature  of  Cathode 
FIG.  8. 

has  no  appreciable  influence  on  the  discharge.  (The  influence 
of  gas  will  be  considered  in  a  later  chapter.)  Care  must  be 
taken  that  the  voltage  applied  between  the  filament  and  the 
anode  is  so  high  that  any  further  increase  in  the  voltage  does  not 
appreciably  increase  the  current.  The  current  obtained  under 
these  conditions  is  then  the  saturation  current  given  by  equation 
(7) .  If  the  current  is  observed  for  different  values  of  the  filament 
temperature  a  curve  is  obtained  such  as  that  shown  in  Fig.  8. 
In  order  to  evaluate  the  constant  6  or  0  we  can  take  logarithms 
of  equation  (7).  Thus: 


logio  I,—\  logio  T = logio  A  - 


.43436 


(10) 


34  THERMIONIC  VACUUM  TUBE 

By  plotting  the  expression  on  the  left-hand  side  against       „ 

a  straight  line  is  obtained  the  slope  of  which  gives  6.  We  shall 
have  occasion  to  return  to  this  equation  when  we  come  to  con- 
sider efficiency  problems  connected  with  the  thermionic  vacuum 
tube. 

It  may  be  remarked  here  that  the  constant  b  can  also  be 
determined  by  photo-electric  means.  The  relation  between 
photo-electric  and  thermionic  phenomena  will  become  apparent 
when  we  come  to  consider  the  photo-electric  effect. 

20.  Influence  of  Surface  Conditions  on  Electron  Affinity. 
By  applying  the  theory  of  images  Debije  l  has  shown  that  it  is 
easier  for  an  electron  to  escape  from  a  sharp  point  than  from  a 
smooth  flat  surface.  The  theory  of  images  involves  a  purely 
mathematical  process  that  tells  us  little  or  nothing  about  the 
nature  of  the  processes  going  on  when  a  electron  escapes  from  a 
surface  and  can  be  applied  only  to  that  part  of  the  process  when 
the  electron  is  co  far  away  from  the  surface  that  molecular  irregu- 
larities in  the  surface  can  be  neglected.  We  can  nevertheless 
obtain  an  indication  of  the  manner  in  which  the  configuration  of 
the  surface  affects  the  electron  emission,  if  we  comply  with  the 
conditions  that  govern  the  application  of  the  theory  of  images. 
This  theory  states  that  the  force  of  attraction  between  a  charged 
body  and  a  conductor  can  be  determined  by  assuming  that  the 
force  is  the  same  as  if  the  conductor  were  replaced  by  another 
charge  which  is,  in  respect  to  size,  shape  and  position,  the  optical 
image  of  the  first  charge  reflected  in  the  surface  of  the  conductor, 
but  is  of  opposite  sign.  Thus  a  charge  —e  at  a  distance  x  from  a 
plane  surface  would  produce  an  image  -\-e  at  a  distance  x  behind 
the  surface.  The  force  of  attraction  between  the  charge  —  e 

—  e2 
and  the  plane  surface  is  therefore  — T  and  the  work  that  must 

be  done  to  remove  the  charge  from  a  distance  XQ  from  the  surface 
to  infinity  is : 

re2  e2 

4-^=4^'  •'•.••• 

which,  when  expressed  in  equivalent  volts  becomes: 

300e 
^1=W 

1  P.  DEBIJE,  Ann.  d.  Phys.,  Vol.  32,  p.  465,  1910. 


ELECTRONS  FROM  SOLID  SUBSTANCES  35 

the  condition  being  that  the  distance  XQ  is  largo  compared  with 
molecular  dimensions. 

The  quantity  w\  does  not  represent  the  total  amount  of 
work  which  an  electron  must  do  to  escape  from  a  plane  surface. 
There  is  still  to  be  added  the  work  W2  done  in  moving  from  the 
interior  of  the  conductor  through  the  interface  and  up  to  the  point 
distant  XQ  from  it.  Schottky  l  and  Langmuir  2  have  made  cer- 
tain assumptions  regarding  the  force  of  attraction  within  this 
region  which  lead  to  the  result  that  the  work  W2  is  equal  to  wi, 

e2 

so  that  the  total  amount  of  work  done  is  =-— .    Since  the  nature  of 

2x0 

the  force  very  close  to  and  inside  the  surface  is  not  known  and 


FIG.  9. 


probably  depends  very  materially  on  the  molecular  structure  of 
the  material  of  the  conductor,3  we  shall  confine  our  considerations 
to  the  force  at  distances  which  are  large  compared  with  the 
molecular  diameters,  and  proceed  to  compute  the  corresponding 
part  MI, -of  the  work  which  an  electron  must  do  to  escape  from  a 
curved  surface  of  radius  r. 

Let  the  surface  be  convex  toward  the  electron  (Fig.  9).  Let 
the  electron  —e  be  at  a  distance  a  and  its  image  +e\  at  a  distance 
a\  from  the  center  of  curvature  of  the  surface.  Then 


=  r2, 


(12) 


1 W.  SCHOTTKY,  Phys.  Zeitsch.,  Vol.  15,  p.  872,  191*. 

2  I.  LANGMUIR,  Trans.  Am.  Electrochem.  Soc.,  Vol.  29,  p.  163,  1916. 

3  J.  FRENCKEL,  Phil.  Mag.,  Vol.  33,  p.  297,  1917. 


36  THERMIONIC  VACUUM  TUBE 

Now  the  force  of  attraction  between  —  e  and  -f-ei  is: 


From  equations  (12)  and  the  geometry  of  the  system,  we  have,  if 
x  be  the  distance  of  the  electron  from  the  surface  : 


Substituting  these  values  for  'e\,  a  and  ai  in  (13)  the  force  of 
attraction  becomes: 


This  equation  holds  for  values  of  x  greater  than  XQ,  where  XQ 
is  large  compared  with  molecular  dimensions.  The  work  which  an 
electron  must  do  to  move  away  from  the  point  XQ  is  obtained  by 
integrating  equation  (14)  between  the  limits  X  =  XQ  and  x=oo. 
The  integration  gives: 


If  we  take  the  radius  of  curvature  of  the  surface  so  large  that 
XQ  is  small  compared  with  it  we  can  write  equation  (15)  in  the 
form: 


The  equivalent  potential  in  volts  is: 


This  shows  that  the  work  of  escape  of  an  electron  from  a  curved 
surface  of  radius  r  is  less  than  that  from  a  plane  surface  by  an 

e2 

amount  equal  to  —  .     It  also  shows  that  if  the  surface  is  irregular 
or 


ELECTRONS  FROM  SOLID  SUBSTANCES  311 

contact  potential  differences  must  exist  between  the  protru-* 
sions  and  the  hollows.  These  potential  differences  are  very 
small,  but  if  irregularities  are  close  to  one  another  the  resulting 
electrostatic  fields  may  be  very  large.  Thus,  if  we  consider  a 
protrusion  and  a  hollow  adjacent  to  it,  each  being  regarded  as 
spherical  surfaces  having  a  radius  of  curvature  of  the  order  of  10 ~6 
cm.,  it  follows  that  the  electrostatic  field  tending  to  drive  electrons 
from  the  protrusion  to  the  hollow  may  be  of  the  order  of  several 
thousands  volts  per  centimeter.  This  would  necessitate  very 
high  plate  potentials  to  overcome  these  fields  and  pull  all  the 
emitted  electrons  over  to  the  anode. 

This  result  is  obtained  by  the  simple  application  of  the  theory 
of  images,  which  unfortunately  does  not  tell  us  much  about  the 
physical  processes  involved.  A  similar  effect  is  to  be  expected 
when  the  surface  of  the  cathode  contains  impurities  having 
electron  affinities  which  are  different  from  that  of  the  material 
of  the  cathode  itself.  Langmuir  l  has  ascribed  the  lack  of  satu- 
ration shown  by  tungsten  filaments  contaminated  with  thorium 
to  the  local  fields  at  the  surface  of  the  filament,  due  to  the  differ- 
ence between  the  electron  affinities  of  thorium  and  tungsten. 
When  the  surface  of  the  filament  consisted  either  of  pure  tungsten 
or  pure  thorium  the  saturation  curve  was  substantially  parallel 
to  the  voltage  axis.  But  when  the  surface  was  a  mixture  of  tung- 
sten and  thorium  the  thermionic  current  continually  increased 
with  the  applied  voltage. 

The  oxide-coated  cathode  is  an  example  of  a  cathode  which 
generally  has  an  irregular  surface.  It  is  obtained  by  coating  a 
platinum  wire  or  ribbon  2  with  oxides  of  the  alkaline  earths.  The 
coated  filament  has  a  much  lower  electron  affinity  and  therefore 
a  higher  thermionic  efficiency  than  the  metals  used  as  sources  of 
thermionic  current.  Its  surface,  however,  is  rough  and  possibly 
is  not  uniformly  active  thermionically.  These  filaments  do  not 
give  such  well-defined  saturation  currents  as  metallic  filaments 
do.  Lack  of  well-defined  saturation  currents  is  generally  not  a 
disadvantage  in  thermionic  tubes.  It  is,  in  fact,  sometimes  a 
distinct  advantage,  as  will  become  evident  from  the  considerations 
given  in  the  following  chapters. 

1  Paper  read  at  Chicago  meeting  of  Am.  Phys.  Soc.,  December,  1919. 

2  A.  WEHNELT,  Ann.  d.  Phys.,  Vol.  4,  425,  1904.     NICOLSON  &  HULL,  U.  S. 
Pat.  1209324,  Brit.  Pat.  17580,  1915. 


38  THERMIONIC  VACUUM  TUBE 

21.  Photo-electric  Effect.  The  process  of  the  dislodgment  of 
electrons  from  solid  bodies  by  means  of  electromagnetic  radiation 
brings  into  play  the  same  forces  that  attend  the  emission  of 
electrons  from  hot  bodies,  and  furnishes  valuable  evidence  to  show 
the  generality  of  that  characteristic  constant  of  solids  which  plays 
such  an  important  part  in  the  operation  of  thermionic  vacuum 
tubes,  namely,  the  electron  affinity. 

In  1887  Hertz  observed  that  when  a  spark  gap  was  illuminated 
with  ultra-violet  light  the  discharge  passed  more  readily  than 
when  the  electrodes  were  in  the  dark.  Soon  after  this  Hallwachs 
discovered  that  the  incidence  of  ultra-violet  light  on  a  zinc  plate 
caused  it  to  become  charged  positively,  or  when  the  plate  was 
first  charged  to  a  negative  potential  and  then  insulated  it  lost  its 
negative  charge  when  exposed  to  the  light.  This  has  since  been 
found  to  be  a  general  property  of  all  conductors  and  could  be  ex- 
plained in  the  light  of  the  electron  theory.  The  energy  of  the  light 
wave  striking  the  substance  stimulates  the  electrons  in  the  atoms 
of  the  substance.  They  thus  acquire  sufficient  energy  to  overcome 
the  force  of  attraction  at  the  surface  of  the  substance  and  escape 
with  a  velocity  which  depends  upon  the  energy  in  the  light  wave 
and  the  amount  of  energy  they  must  expend  to  overcome  the 
surface  force.  Thus,  if  the  amount  of  energy  acquired  by  the 
electron  in  the  substance  from  the  light  is  W  and  the  work  which 
the  electron  must  do  to  overcome  the  surface  force  is  w,  then  it 
escapes  from  the  substance  with  a  kinetic  energy. 


(18) 


where  v  is  the  velocity  of  escape  and  m  the  mass  of  the  electron. 
It  will  be  understood  that  of  the  electrons  in  the  substance  those 
that  happen  to  be  near  to  the  surface  have  to  overcome  only  the 
surface  force,  while  those  that  are  further  in  the  interior  will  have 
to  do  an  extra  amount  of  work  in  forcing  their  way  out.  We 
can  therefore  expect  electrons  to  be  emitted  by  the  light  with 
velocities  ranging  from  zero  to  a  definite  maximum  value.  This 
maximum  value  expressed  in  volts  is  the  electron  affinity.  We 
shall  in  the  following  consider  only  those  electrons  that  have 
this  maximum  velocity,  and  equation  (18)  will  be  understood  to 
refer  to  these  maximum  values.  If  a  plate  is  placed  in  front 
of  the  electron  emitting  substance  or  cathode,  the  emitted  electrons 
can  be  driven  back  to  the  cathode  by  the  application  of  a  potential 


ELECTRONS  FROM  SOLID  SUBSTANCES 


difference  between  it  and  the  anode.  If  the  electron  emerges  from 
the  surface  with  a  velocity  v  it  will  be  capable  of  moving  against 
,an  electric  field  until  it  has  spent  its  kinetic  energy  ^mv2,  where 
m  is  the  mass  of  the  electron.  If  the  maximum  voltage  against 
which  the  electron  can  move  in  virtue  of  its  own  kinetic  energy 
is  V,  then  Ve  =  ^mv2.  The  velocity  of  an  electron  is  commonly 
expressed  in  terms  of  the  voltage  V,  instead  of  centimeters  per 
second.  Equation  (18)  can  then  be  written 


(19) 


where  e  is  the  electronic  charge. 

This  voltage  can  be  determined  with  the  arrangement  shown 
in  Fig.  10.  A  is  the  photo-electric  cathode  which  can  be  illu- 
minated with  ultra-violet  light,  and  B  is  the  anode.  By  means,  of 


-wwvwvwvwww 

t 


FIG.  10. 

the  potentiometer  shown  the  voltage  between  A  and  B  can  be 
adjusted  to  any  desired  value  tending  to  drive  the  electrons  in  the 
direction  B  to  A.  Unless  this  voltage  exceeds  a  definite  amount 
the  electrons  emitted  from  A  under  the  influence  of  the  light  will 
travel  all  the  way  across  to  B  in  virtue  of  the  velocity  with  which 
they  are  emitted,  and  the  resulting  current  established  in  the 
circuit  can  be  measured  with  a  current-measuring  device. 
If  we  now  measure  the  current  for  increasing  values  of  the  voltage 
V,  the  current  decreases  until  the  voltage  is  large  enough  to  return 
all  the  emitted  electrons  to  the  cathode  before  they  can  reach  the 
anode.  By  plotting  the  photo-current  against  the  voltage  V 
a  curve  is  obtained  such  as  that  shown  in  Fig.  11.  The  voltage 
is  reckoned  negative  when  the  receiving  plate  B  is  negative  with 


40 


THERMIONIC  VACUUM  TUBE 


respect  to  the  emitting  plate  A.  The  point  at  which  the  curve 
cuts  the  voltage  axis  gives  the  maximum  velocity  with  which  the 
electrons  are  emitted  from  the  cathode. 

Experiment  has  shown  the  remarkable  result  that  the  maximum 
velocity  of  emission  is  independent  of  the  temperature  of  the 
cathode.  The  maximum  velocity  of  photo-electric  emission  is 
furthermore  independent  of  the  intensity  of  the  light  with  which 
the  cathode  is  illuminated.  If  the  intensity  of  the  light  is  in- 
creased only  the  number  of  electrons  emitted  increases  but  their 
velocity  stays  the  same,  provided  that  by  changing  the  intensity 
of  the  light  we  do  not  at  the  same  time  change  its  quality,  that 


Anode    Potenfial 
FIG.  11. 

is,  its  wave  length  distribution.  The  frequency_of__the  incident 
light  is  the  only  factor  that  influences  the  Telocity  of  emission 
when  dealing  with  one  substance.  For  the  same  light  frequency 
and  different  substances,  the  emission  velocity  depends  upon 
the  electron  affinity  of  the  substances.  Millikan  1  has  shown 
that  if  the  maximum  voltage  necessary  to  keep  the  emitted  elec- 
trons from  reaching  the  anode  be  plotted  against  the  frequency 
of  the  light  the  linear  relation  shown  in  Fig.  12  is  obtained. 
Thus: 


(20) 


where  h  is  a  constant  and  v  the  frequency  of  the  light  falling 

on  the  cathode.     Referring  to  equations  (18)  and  (19)  it  is  seen 

1  R.  A.  MILLIKAN,  Phys.  Rev.,  Vol.  7,  p.  355,  1916. 


ELECTRONS  FROM  SOLID  SUBSTANCES 


41 


that  W,  the  energy  acquired  from  the  light  by  the  electron  in  the 

This 


w 


substance  is  equal  to  hv,  and  the  constant  C  is  equal  to  — . 

e 

extremely  important  experimental  result  shows  that  light  energy 
can  be  expressed  by  the  product  of  the  frequency  of  the  light  and  a 
constant.  Indeed,  Millikan  found  that  this  constant  is  the  same 
as  Planck's  constant  of  action.  Furthermore,  C  has  actually  been 
found  to  be  equal  to  the  electron  affinity  <£.  We  therefore  have 
as  the  fundamental  photo-electric  equation: 


Ye  =  hv — <f)e.    .     .     .'    .     .     .     .     (21) 

This  equation  was  originally  deduced  theoretically  by  Einstein  1 
on  an  assumption  that  he  has  since  abandoned.    But  Millikan's  ex- 


Frequency  of  Incident  Light 


FIG.  12. 

periments  have  shown  that  this  equation  holds  with  a  high  degree 
of  accuracy  and  have  placed  beyond  doubt  the  correctness  of  this 
very  simple  expression  for  the  light  energy  necessary  to  dislodge 
an  electron. 

So  far  we  have  considered  only  the  voltage  applied  between 
the  plates  A  and  B.  This  is,  however,  not  the  only  voltage  that 
affects  the  motion  of  the  electrons.  There  remains  to  be  con- 
sidered the  contact  potential  difference  between  the  plates,  which 
•is  equal  to  the  difference  between  the  electron  affinities  of  the  two 
plates.  This  potential  difference,  which  can  be  measured  by  the 
method  explained  in  Section  18,  must  be  added  to  the  applied 
voltage. 

1  A.  EINSTEIN,  Ann.  d.  Phys.  (4)  Vol.  20,  p.  199,  1905. 


42  THERMIONIC  VACUUM  TUBE 

Referring  now  to  Fig.  12,  it  is  seen  that  there  is  a  definite 
frequency  VQ  of  the  light  at  which  the  voltage  necessaiy  to  drive 
the  electrons  back  becomes  zero.  This  means  that  for  frequencies 
lower  than  this  value  no  electrons  escape  at  all.  Putting  7  =  0 
in  equation  (21)  we  get: 


(22) 


This  limiting  frequency,  commonly  referred  to  as  the  photo- 
electric long  wavelength  limit  is  a  fundamental  property  of  solids, 
and,  is  equal  to  a  constant  multiplied  by  the  electron  affinity  — 
the  same  constant  that  plays  such  an  important  part  in  the  emission 
of  electrons  from  hot  filaments  in  the  thermionic  vacuum  tube. 

The  quantities  h  and  e  are  universal  constants,  that  is,  their 
values  are  independent  of  the  matter  under  investigation  and  the 
conditions  of  the  experiments.  Their  values  are  /i  =  6.55XlO~27 
erg.  sec.,  and  e  =  4.77XK)-10  E.S.  units. 

22.  Control  of  Space  Current  by  Means  of  an  Auxiliary  or 
Third  Electrode.  A  convenient  and  what  has  proved  to  be  a  very- 
valuable  means  of  varying  a  space  current  is  obtained  by  placing 
a  third  electrode  in  the  neighborhood  of  the  cathode  and  applying 
potential  variations  to  it.  This  scheme  was  used  by  de  Forest 
to  control  the  electron  discharge  in  his  audion  detector  in  1907.1 
He  later  gave  the  auxiliary  electrode  the  form  of  a  wire  gauze  or 
grid  placed  in  the  path  of  the  discharge  between  cathode  and 
anode.2  About  the  same  time  von  Baeyer  3  also  used  an  auxiliary 
electrode  to  control  a  thermionic  current  from  a  hot  filament. 
In  von  Baeyer's  arrangement  the  anode  was  a  cylinder  and  the 
cathode  a  wire  placed  along  its  axis.  The  third  electrode  was  a 
wire  gauze  bent  into  the  form  of  a  cylinder  and  placed  between 
cathode  and  anode.  A  similar  scheme  was  also  used  by  Lenard  4 
in  1902  in  connection  with  photo-e^ctric  experiments.  It  is 
hardly  necessary  to  say  that  the  insertion  of  the  grid  has  made  the 
audion  a  device  of  immense  practical  importance  and  enabled  it 
to  perform  functions  that  would  otherwise  have  been  impossible. 

1  LEE  DE  FOREST,  U.  S.  Patent  No.  841387,  1907. 

2  LEE  DE  FOREST,  U.  S.  Patent  No.  879532,  1908. 

3  O.  VON  BAEYER,  Verb.  d.  D.  Phys.  Ges.,  Vol.  7,  p.  109,  1908. 

4  P.  LENARD,  Ann.  d.  Phys.,  Vol.  8,  p.  149,  1902. 


ELECTRONS  FROM  SOLID  SUBSTANCES 


43 


The  quantitative  effect  of  the  third  electrode  was  first  given 
by  the  author.1  The  nature  of  this  effect  can  be  understood 
from  the  following:  In  Fig.  13  F  is  the  cathode,  P  the  anode 
and  G  the  auxiliary  electrode,  which  may  be  in  the  form  of  a  wire 
grid  or  gauze.  The  battery  Ep  maintains  the  anode  at  a  positive 
potential  with  respect  to  F,  while  G  can  be  given  any  desired 
negative  potential  by  means  of  the  battery  Eg.  The  positive 
potential  on  P  has  the  effect  of  drawing  the  electrons  through  the 
grid  to  P,  whereas  the  negative  potential  on  the  grid  tends  to 
drive  them  back  to  the  cathode,  and  by  increasing  Eg  a  value 
can  be  reached  for  which  all  the  emitted  electrons  are  returned 
to  the  cathode.  Eg  therefore  takes  the  place  of  V  in  the  arrange- 

6 


i 


FIG.  13. 

ment  shown  in  Fig.  10.  But  there  is  this  difference  that  while  in 
Fig.  10  the  electric  field  between  the  plates  is  due  only  to  applied 
voltage  V  and  the  contact  potential  difference  between  the  plates, 
in  the  present  case  there  is  a  third  factor  which  influences  the  field 
between  the  cathode  and  grid,  namely,  the  potential  difference 
due  to  the  battery  Ep.  Thus,  if  Ett  is  zero  and  the  contact  potential 
difference  between  F  and  GP  be  supposed  for  the  present  to  be 
also  zero,  then  the  electric  field  between  F  and  G  is  not  zero  but 
has  a  definite  value  depending  upon  the  structural  parameters  of 
the  device  and  the  potential  difference  between  P  on  the  one  hand 
and  F  and  G  on  the  other.  (F  and  G  are  now  supposed  to  be 
metallically  connected.)  This  is  due  to  the  fact  that  the  poten- 
tial of  P  causes  a  stray  field  to  act  through  the  openings  of  the 
grid.  If  the  potential  difference  between  P  and  FG  be  equal  to 
1  H.  J.  VAN  DER  BIJL,  Verb.  d.  D.  Phys.  Ges.,  Vol.  15,  p.  338,  1913. 


44  THERMIONIC  VACUUM  TUBE 

EP  the  field  at  a  point  near  F  is  equivalent  to  the  field  that  would 

ET 

be  sustained  at  that  point  if  a  potential  difference  equal  to  -  -  be 

applied  directly  between  the  cathode  A  and  a  plane  coincident 
with  that  of  the  grid.  For  the  usual  connection  in  which  the 
plate  P  is  positive  the  direction  of  this  field  is  such  as  to  draw 
electrons  away  from  the  cathode.  But  it  does  not  draw  the  elec- 
trons to  the  grid,  as  would  be  the  case  if  a  potential  difference  were 
applied  directly  between  the  grid  and  cathode;  it  tends  to  draw 
electrons  to  the  anode  through  the  openings  of  the  grid. 

Besides  this  stray  field  there  is  also  the  contact  potential 
difference  K  between  F  and  GP.  Hence,  if  K  be  reckoned  positive 
when  it  tends  to  draw  electrons  away  from  the  cathode,  and  if  the 
maximum  velocity  of  emission,  expressed  in  volts,  of  the  electrons 
liberated  from  the  cathode  be  V,  then  in  order  to  drive  all  the 
emitted  electrons  back  to  the  cathode  we  must  apply  between 
cathode  and  grid  a  potential  difference  equal  to 

E 

TJT    V\,  /OQ"\ 

^*-7+€> <23) 

where  e=K+V.  This  expression  can  be  regarded  as  the  effective 
voltage  when  the  potential  difference  between  grid  and  cathode 
is  zero.  If  this  potential  difference  be  made  equal  to  Eg  the 
effective  voltage  becomes 

Ep  '  TO  '-€ (24) 


In  this  expression  Ep  and  Eg  are  the  potentials  of  the  anode  and 
grid  with  respect  to  that  of  the  cathode,  which  can  be  regarded 
as  the  zero  of  potential.  Hence,  when  Eg  is  varied  the  potential 
difference  between  grid  and  plate  also  changes.  When  I  first 
established  this  linear  stray  field  relation  in  1913,  I  expressed 
the  result  by  the  equation 


(25) 


where  v  is  the  potential  difference  between  cathode  and  grid  and  V 
that  between  grid  and  anode.     I  also  stated  that  A;  is  a  constant 


ELECTRONS  FROM  SOLID  SUBSTANCES 


45 


depending  on  the  grid  and  d  the  distance  between  grid  and  anode. 
In  testing  this  relation  the  grid  remained  grounded  while  the 
potential  of  the  cathode  was  varied.  This  made  it  possible  to 
keep  the  potential  difference  V  between  grid  and  anode  nearly 
constant  while  varying  the  potential  difference  between  cathode 
and  grid.  The  accuracy  with  which  equation  (25)  was  found 
to  hold  is  shown  by  Fig.  14. 1  In  the  case  of  the  lower  curve  the 


7 


2Q  40  60  80  10 

Anode  -Grid  Voltcige 

FIG.  14 


distance  d  between  grid  and  anode  was  6.7  mm.,  while  the 
upper  curve  was  obtained  with  d  =  2.5  mm.  If,  instead  of  plot- 
ting V  as  abscissae,  we  plot  — ,  the  two  curves  coincide. 

rfj 

It  can  readily  be  seen  that  if  we  substitute  —  Ea  for  v  and  EP, 
the  potential  of  the  anode  with  respect  to  the  cathode  for  V,  the 

1  Loc.  cit.,  p.  339. 


46  THERMIONIC  VACUUM  TUBE 

potential  of  the  anode  with  respect  to  the  grid,  then  (24)  and  (25) 
give  the  same  result  provided  that 


and 

£  =  /*-! (26) 

I  have  since  verified  this  relationship  between  ^  and  the 
structural  parameters  of  the  tube  on  the  basis  of  an  extensive  series 
of  experiments  carried  out  in  the  research  laboratories  of  the 
Western  Electric  Company,  and  have  also  expressed  k  in  terms  cf 
the  mesh  of  the  grid  and  the  diameter  of  the  grid  wires.  (See 
Chapter  VII,  p.  231.) 

The  constant  n  is  a  very  important  constant  of  the  three- 
electrode  tube  and,  as  will  be  shown  later,  expresses  the  maximum 
voltage  amplification  obtainable. 

Expression  (24)  or  (25)  can  be  regarded  as  the  fundamental 
relationship  of  the  three-electrode  vacuum  tube.  The  current  in 
the  circuit  FPA  (Fig.  13)  is  obviously  a  function  of  the  expression 
(24).  Hence,  if  the  potential  of  the  cathode  be  maintained 
constant,  the  fundamental  expression  for  the  current  in  a  three- 
electrode  tube  is 


(26) 


where  Ep  and  E</  are  the  potentials  of  the  anode  and  grid  with 
respect  to  that  of  the  cathode.  We  shall  have  occasion  to  make 
extensive  use  of  this  relationship  in  dealing  with  the  three- 
electrode  thermionic  tube. 

It  will  be  shown  later  that  a  device  like  that  shown  in  Fig.  13 
and  whose  current-voltage  characteristic  can  be  expressed  by 
the  function  (26)  can  be  used  as  amplifier,  radio  detector,  oscilla- 
tion generator,  etc. 

A  device  which  depends  for  its  current  on  the  emission  of 
electrons  by  photo-electric  means  is  not  as  suitable  for  these 
purposes  as  thermionic  devices,  because  photo-electric  currents 
are  generally  very  small  and  the  emission  of  electrons  by  heat 
is  much  more  practical  than  emission  under  the  influence  of  light. 


ELECTRONS  FROM  SOLID  SUBSTANCES  47 

23.  Secondary   Electron   Emission.    Delta    Rays.    We   now 

come  to  a  consideration  of  the  third  agency  whereby  electrons 
can  be  dislodged  from  substances,  viz.,  by  the  impact  of 
electrons. 

When  electrons  are  made  to  impinge  on  a  metallic  plate  they 
give  rise  to  the  radiation  of  X-rays.  These  X-rays  comprise  a 
general  X-radiation  superimposed  upon  which  there  is  under 
certain  conditions  a  so-called  characteristic  radiation.  The  same 
effects  are  produced  when  the  plate  is  exposed  to  X-rays.  The 
frequency  of  the  characteristic  radiation  is  closely  related  to  the 
atomic  properties  of  the  substance  from  which  it  originates,  and 
is  not  produced  unless  the  velocity  of  the  impinging  electrons, 
which  we  shall  call  the  primary  electrons,  exceeds  a  certain  definite 
value.  When  the  characteristic  radiation  is  produced  there  is 
also  a  copious  emission  of  electrons  from  the  plate.  These  elec- 
trons are,  of  course,  returned  to  the  plate  by  the  potential  differ- 
ence between  it  and  the  cathode  from  which  the  primary  electrons 
proceed,  and  in  order  to  measure  them  a  special  circuit  arrange- 
ment, such  as  will  be  described  below  must  be  used.  Every 
metal  possesses  a  number  of  characteristic  frequencies  and  although 
in  general  X-rays  have  so  far  been  investigated  extensively  only 
at  high  frequencies — at  voltages  corresponding  to  several  thousand 
volts,  they  are  also  produced  by  low  velocity  electrons.  Dember  1 
has,  for  example,  produced  X-rays  and  measured  their  effect  at  a 
voltage  as  low  as  17  volts.  These  rays  are  so  soft  that  they  can- 
not penetrate  the  walls  of  the  containing  vessel.  Their  effects 
must  therefore  be  studied  inside  the  vessel.  Likewise  in  the  case 
of  the  secondary  electron  emission  it  is  not  necessary  for  the  pri- 
mary electrons  to  strike  the  electrode  with  high  velocities. 

The  impact  voltage  at  which  secondary  electrons  are  emitted 
depends  on  the  nature  of  the  surface  of  the  emitting  electrode. 
The  phenomenon  of  secondary  electron  emission,  which  is  some- 
times referred  to  as  "  Delta  rays  "  shows  this  important  property 
that  as  the  velocity  of  the  primary  electrons  is  increased  the  num- 
ber of  secondary  electrons  emitted  per  impinging  electron  increases. 
In  fact,  if  the  applied  voltage,  i.e.,  the  velocity  with  which  the 
primary  electrons  strike  the  emitting  electrode,  be  increased  to  a 
sufficiently  high  value  one  primary  electron  can  expel  as  many  as 
twenty  secondary  electrons. 

1 H.  DEMBER,  Verb,  d.  D,  Phys,  Ges.,  Vol.  15,  p.  560,  1913. 


48 


THERMIONIC  VACUUM  TUBE 


The  presence  of  secondary  electrons  can  easily  be  demon- 
strated by  means  of  the  circuit  arrangement  shown  in  Fig.  15. 
The  plate  P  is  kept  at  a  constant  positive  potential  with  respect  to 
the  filament  F  by  the  battery  E.  When  no  potential  difference 
exists  between  filament  and  grid  G,  the  current  in  the  circuit  FGA 
is  very  small,  because  practically  all  the  electrons  emitted  from  the 
filament  are  drawn  through  the  openings  of  the  grid  and  thrown 
on  to  the  plate.  If  now  the  potential  of  the  grid  (positive  with 
respect  to  the  filament)  be  increased  the  current  to  the  grid  l  at 
first  increases,  as  shown  by  the  part  OA  in  the  curve  in  Fig.  16. 
When  the  grid  potential  reaches  a  certain  value  the  current,  as 
indicated  by  the  ammeter  A  (Fig.  15)  begins  to  decrease,  drops 


FIG.  15. 


FIG.  16. 


down  to  zero  at  B  and  then  flows  in  the  reversed  direction.  The 
explanation  for  this  is  that  while  the  potential  difference  between 
filament  and  grid  is  small,  the  electrons  that  strike  the  grid  enter 
it,  but  as  the  positive  grid  potential  is  increased  the  electrons  on 
striking  the  grid  emit  so-called  "  secondary  electrons  "  from  it 
and  these  are  drawn  to  the  plate  which  is  maintained  by  the  bat- 
tery E  at  a  positive  potential  with  respect  to  the  grid.  The  net 
current  as  indicated  by  the  ammeter  A  is  the  sum  of  electrons 
entering  the  grid  and  those  leaving  it.  When  the  velocity  with 
which  the  electrons  strike  the  grid  increases  beyond  a  certain 
value,  one  primary  electron  can  knock  out  more  than  one  second- 
ary electron  from  the  grid,  and  the  current  in  the  circuit  FGA 
reverses.  When  the  positive  grid  potential  is  increased  to  such  an 

1  The  direction  of  current  is  here,  as  throughout  the  following,  taken  to 
the  direction  in  which  electrons  move. 


ELECTRONS  FROM  SOLID  SUBSTANCES  49 

extent  that  the  grid  becomes  positive  with  respect  to  the  plate, 
the  secondary  electrons  are  no  longer  drawn  away  to  the  plate, 
but  are  driven  back  to  the  grid  so  that  the  reversed  current  in  the 
grid  circuit  again  decreases,  as  shown  at  C  (Fig.  16),  and  finally 
assumes  the  original  direction.1 

Considering  now  the  current  as  indicated  by  the  ammeter  A 
and  the  voltage  between  filament  and  grid,  we  see  that  over  the 
region  ABC  (Fig.  16),  the  current  decreases  as  the  voltage  increases. 
The  part  of  the  curve  ABC  therefore  represents  a  negative  resist- 
ance characteristic.  It  will  be  shown  later  that  a  device  which 
possesses  a  negative  resistance  can  function  as  an  amplifier  and  a 
generator  of  continuous  oscillations. 

1  A.  W.  HULL,  Phys.  Rev.,  Vol.  7,  p.  1,  1916;  Proc.  I.  R.  E.,  Vol.  5,  p.  5, 
1918. 


CHAPTER  IV 
PHYSICS  OF  THE  THERMIONIC  VALVE 

24.  Current-voltage  Characteristic  of  the  Thermionic  Valve. 
In  discussing  the  elements  of  thermionics  we  considered  only  the 
saturation  current,  that  is,  the  current  obtained  from  a  cathode 
at  any  desired  temperature  by  applying  a  voltage  which  is  so  high 
that  all  the  electrons  emitted  from  the  cathode  are  drawn  away 
to  the  anode.  For  all  values  of  the  applied  voltage  greater  than 
the  minimum  value  necessary  to  secure  this,  the  current  is  inde- 
pendent of  the  voltage.1  The  saturation  current  is  important 
in  telling  us  what  the  maximum  current  is  that  can  be  obtained 
from  a  cathode  of  given  area  at  a  given  temperature.  In  operating 
thermionic  devices  as  amplifiers,  detectors,  etc.,  we  make  use  of 
the  variation  in  current  with  variation  in  applied  voltage.  The 
conditions  under  which  the  saturation  current  is  obtained  are, 
therefore,  unsuitable  for  these  purposes.  In  this  chapter  will  be 
discussed  the .  phenomena  encountered  when  the  applied  voltage 
is  not  high  enough  to  draw  all  the  emitted  electrons  away  to  the 
anode  as  fast  as  they  are  emitted  from  the  cathode. 

Let  us  consider  a  simple  thermionic  device  consisting  of  a 
cathode  which  can  be  heated  to  any  desired  temperature  and  an 
anode  placed  at  a  convenient  distance  from  it  both  being  placed  in 
a  vessel  which  is  evacuated  to  such  an  extent  that  the  residual 
gas  plays  no  appreciable  part  in  the  current  convection  between 
cathode  and  anode.  We  shall  for  the  present  suppose  that  the 
cathode  is  a  plane  equipotential  surface,  and  the  anode  a  plane 
parallel  to  the  cathode.  If  the  cathode  be  maintained  at  a  definite 
temperature  T\  and  the  current  to  the  anode  be  observed  for 

1  In  practice  it  is  found  that  this  is  seldom  strictly  true.  The  current 
usually  increases  somewhat  with  the  voltage,  but  not  nearly  as  fast  as  for 
the  lower  voltages.  If  the  applied  voltage  is  raised  to  excessive  values  gas 
can  be  liberated  from  the  electrodes  and  then  the  current  may  again  increase 
rapidly  with  increase  in  the  plate  voltage.  (See  Chapter  V.) 

50 


PHYSICS  OF  THE  THERMIONIC  VALVE 


51 


various  values  of  the  voltage  applied  between  the  cathode  and 
anode,  a  curve  OA\  of  Fig.  17  is  obtained.  Any  increase  in  the 
voltage  beyond  the  value  given  by  AI  causes  practically  no  further 
increase  in  the  current  and  we  get  the  part  A\B\.  In  practice 


Anode   Vol+otge 

FIG.  17. 

this  line  is  seldom  horizontal  but  usually  slopes  upward.  This 
part  of  the  curve  corresponds  to  the  condition  when  all  the  emitted 
electrons  are  drawn  to  the  anode  as  fast  as  they  are  emitted  from 
the  cathode.  The  corresponding  current  is,  therefore,  the  satura- 

_  k-' 


Cod" node  Temperature 

FIG.  18. 


tion  current.  If  the  temperature  of  the  cathode  be  increased  to 
T2  the  curve  OA2B2  will  be  obtained.  If  these  values  of  the 
saturation  current  be  plotted  against  the  corresponding  tempera- 
tures of  the  cathode,  the  curve  shown  in  Fig.  18  will  be  obtained, 


52  THERMIONIC  VACUUM  TUBE 

the*  relation  being  given  by  Richardson's  equation  (equation  (7), 
Chapter  III). 

The  fact  that  a  finite,  and  sometimes  a  considerable,  voltage  is 
needed  to  obtain  the  saturation  current,  shows  that  the  current 
is  limited  by  some  means  that  is  equivalent  to  a  resistance.  The 
current  through  a  piece  of  wire  is  in  a  sense  limited  because,  if 
the  wire  is  of  finite  length,  i.e.,  if  it  has  a  finite  resistance,  a  finite 
potential  difference  must  be  applied  to  the  ends  of  the  wire  to 
give  a  finite  current.  A  device  which  obeys  Ohm's  law  gives,  of 
course,  a  linear  relation  between  current  and  voltage,  that  is,  when 
the  voltages  and  corresponding  currents  are  plotted  on  rectangular 
coordinates  the  result  is  a  straight  line  passing  through  the  origin, 
and  the  reason  why  the  line  does  not  coincide  with  the  ordinate 
axis  is  because  the  wire  has  a  finite  resistance  which  limits  the 
current.  In  the  passage  of  current  through  a  gas,  the  gas  itself 
resists  the  flow  of  current,  the  resistance  being  partly  due  to  the 
collisions  of  the  carriers  with  the  gas  molecules.  In  a  thermionic 
vacuum  tube,  however,  this  resistive  medium  is  removed,  because 
the  vacuum  is  so  high  that  the  gas  contributes  practically  nothing 
to  the  current  convection  through  the  tube. 

If  there  were  nothing  to  limit  the  current  in  a  vacuum  tube,  the 
current  would  increase  very  rapidly  with  the  applied  voltage. 
This  is,  however,  not  the  case;  there  are  factors  that  have  a  very 
pronounced  influence  in  limiting  the  current  in  a  vacuum  tube  in 
such  a  way  as  to  give  a  characteristic  somewhat  like  that  shown 
in  Fig.  17. 

One  of  these  factors  is  the  repelling  effect  of  electrons  in  the 
space  between  cathode  and  anode  on  other  electrons  coming  from 
the  cathode.  This  is  due  to  the  volume  density  of  electrification 
or  space  charge  of  the  electrons  in  the  space.  Whenever  current 
is  carried  by  dislodged  charges,  their  space  charge  must  be 
taken  into  consideration.  Thus,  in  the  conduction  of  electricity 
through  gases  the  fundamental  equations  always  contain  or  involve 
the  space  charge  equation,  or  Poisson's  equation  which  was  given 
on  page  15,  Chapter  I.  In  conduction  through  gases  we  have 
to  deal  with  the  more  general  case  where  both  positive  and  negative 
charges  are  present.  The  resultant  space  charge  is  the  difference 
between  that  of  the  negative  and  that  of  the  positive  carriers, 
and  since  the  positives  or  negatives  generally  do  not  move  with  the 
same  speed,  the  problem  is  generally  complicated.  In  the  ther- 


PHYSICS  OF  THE  THERMIONIC  VALVE  53 

mionic  vacuum  tube,  on  the  other  hand,  we  have  to  deal  only  with 
electrons,  and  therefore  encounter  in  space  charge  a  specific  and 
comparatively  simple  manifestation  of  a  general  phenomenon 
always  met  with  in  convection  of  current  by  dislodged  charges. 
The  limitation  of  current  by  space  charge  when  the  current  is 
carried  only  by  electrons  was  noticed  in  the  early  experiments  of 
Lenard,  Stoletow  and  von  Schweidler  on  the  photo-electric  effect. 
In  these  experiments  the  effect  was  very  small  on  account  of  the 
smallness  of  the  currents  encountered  in  photo-electric  phenomena. 

In  1907  Soddy  1  made  use  of  the  property  of  metallic  oxides 
discovered  by  Wehnelt 2  in  1904,  namely,  that  they  emit  electrons 
copiously  when  heated.  Soddy  found  that  if  the  vacuum  was  made 
high  by  the  vaporization  of  calcium,  the  thermionic  current  sud- 
denly decreased  to  a  small  fraction  of  its  value  at  the  higher  pres- 
sure. Soddy  thought  that  this  meant  that  Wehnelt  cathodes 
became  inactive  in  very  high  vacuum.  An  explanation  of  what 
Soddy  had  observed  was  given  by  O.  W.  Richardson  and  J.  E. 
Lilienfeld.3  The  latter  pointed  out  more  specifically  that  the 
vacuum  produced  in  Soddy's  experiment  by  the  vaporization  of 
calcium  resulted  in  the  condition  where  the  number  of  electrons 
carrying  the  current  became  large  compared  with  the  number  of 
gas  molecules  in  the  path  of  the  discharge.  The  number  of 
positive  ions  formed  by  collision  ionization  of  the  electrons  with 
the  gas  molecules  became  negligibly  small,  so  that  there  was  es- 
tablished a  negative  space  charge  due  to  the  electrons  and  this 
reduced  the  current  flow  observed  by  Soddy. 

We  shall  see  later  that  the  current  in  thermionic  tubes  is  limited 
not  only  by  space  charge  of  the  electrons  but  by  other  factors  as 
well.  For  the  present  we  shall  consider  only  this  factor,  this  being 
the  simplest  of  the  current  limiting  factors,  and  later  explain  how 
other  effects  contribute  to  give  the  current  voltage  characteristics 
of  the  thermionic  tube. 

The  limitation  of  current  by  space  charge  can  be  demonstrated 
in  a  qualitative  manner  as  follows:  Let  a  definite  voltage  E\  be 
applied  between  cathode  and  anode.  We  shall  for  the  present 
assume  that  the  cathode  is  an  equipotential  surface.  If  now  the 

1  SODDY,  Nature,  Nov.,  1907,  p.  53. 

2  WEHNELT,  Ann.  d.  Phys.,  Vol.  14,  p.  425,  1904. 

3  O.  W.  RICHARDSON,  Nature,  Jan.,  1908,  p.  197;   J.  E.  LILIENFELD,  Phys. 
Zeitschr.,  Vol.  9,  p.  193,  1908 


54  THERMIONIC  VACUUM  TUBE 

current  to  the  anode  be  observed  as  a  function  of  the  temperature 
of  the  cathode,  the  current  will  at  first  increase  until  it  reaches  a 
value  indicated  by  Ci  (Fig.  18).  Any  further  increase  in  cathode 
temperature  causes  no  further  increase  in  the  current,  and  the 
part  CiDi  of  the  curve  is  obtained.  The  current  given  by  C\Di  is 
frequently  referred  to  as  the  "  temperature  saturation  current," 
and  the  condition  characterized  by  this  lack  of  increase  of  current 
with  increase  of  cathode  temperature  as  temperature  saturation. 
The  reason  why  under  thess  conditions  the  current  does  not  in- 
crease along  CiC2Cs  as  would  be  expected  from  Richardson's 
equation  is  because  at  cathode  temperatures  greater  than  that  cor- 
responding to  Ci,  so  many  electrons  are  emitted  that  the  resulting 
volume  density  of  their  charge  causes  all  other  emitted  electrons 
to  be  repelled,  and  these  return  to  the  cathode.  The  applied 
voltage  EI  is  then  not  high  enough  to  draw  all  the  emitted  electrons 
away  to  the  anode.  If  now  the  voltage  be  increased  to  E^,  the 
current  increases,  since  more  electrons  are  now  drawn  away  from 
the  supply  at  the  cathode,  the  full  space  charge  effect  being 
maintained  by  fewer  electrons  being  compelled  to  return  to  the 
cathode.  From  Fig.  18  it  is  seen  that  with  the  voltage  E%  the 
cathode  must  be  raised  to  a  minimum  temperature  corresponding 
to  €2  before  the  full  space  charge  effect  can  manifest  itself.  It  is 
seen,  then,  that  the  higher  the  applied  voltage,  the  higher  must 
be  the  cathode  temperature  to  obtain  the  full  space  charge  effect. 
It  is  also  seen  that  the  part  OC  of  Fig.  18  corresponds  to  the  part 
AB  of  Fig.  17,  and  CD  of  Fig.  18  to  OA  of  Fig.  17.  The  saturation 
current  is  obtained  when  the  applied  voltage  is  so  high  that  a 
variation  of  voltage  does  not  cause  any  appreciable  variation  in 
current,  while  the  condition  under  which  the  thermionic  tube 
operates,  as  a  voltage  operating  device,  is  characterized  by  the 
condition  that  the  cathode  temperature  is  so  high  that  the  current 
does  not  vary  appreciably  with  variation  in  cathode  temperature. 
25.  Current-voltage  Relation  for  Infinite  Parallel  Plates.  To 
get  an  understanding  of  the  quantitative  effect  on  the  current 
by  the  space  charge  of  the  electrons,  it  may  be  well  first  to  consider 
the  ideal  and  simple  case  that  results  when  we  neglect  the  com- 
plicating factors  encountered  in  practice  and  then  consider  the 
modifications  introduced  by  these  factors.  In  deriving  the  equa- 
tions for  this  simple  oase,  we  shall  therefore  assume  that  the  elec- 
trodes are  infinitely  large  parallel  plates,  capable  of  being  main- 


PHYSICS  OF  THE  THERMIONIC  VALVE  55 

tained  at  any  desired  temperature.  Both  electrodes  will  be 
assumed  to  be  equipotential  surfaces.  The  cathode  will  be  main- 
tained at  a  high  temperature,  the  anode  remaining  cold.  It 
follows  from  Richardson's  theory  that  the  hot  plate  will  emit 
electrons,  the  emission  being  the  result  of  the  kinetic  energy  of 
the  electrons  becoming  sufficiently  high  to  overcome  the  surface 
force  that  tends  to  hold  the  electrons  within  the  cathode.  It  will 
be  recognized  that  the  energy  and  distribution  of  energy  of  the 
electrons  play  an  important  part  in  the  mechanism  of  electron 
emission.  A  derivation  of  the  relation  between  current  and 
voltage,  which  takes  into  consideration  the  energy  distribution 
between  the  electrons,  is  quite  complicated.  J.  J.  Thomson 1 
has  given  the  equations  resulting  from  the  assumption  that  the 
electrons  all  emerge  with  one  initial  velocity.  In  1911,  C.  D. 
Child  2  gave  the  full  solution,  based  on  the  assumption  that  the 
initial  velocity  of  emission  is  zero. 

Langmuir3  of  the  General  Electric  Company  and  Schottky4 
also  published  derivations  of  the  space  charge  equation  and  made 
a  careful  investigation  of  some  of  the  phenomena  observed  in 
thermionic  tubes. 

We  shall  now  derive  Child's  equation,  making  the  same  assump- 
tions, and  then  consider  the  modifications  introduced  by  a  con- 
sideration of  the  factors  neglected  in  the  simple  derivation,  and 
more  particularly  how  these  factors  contribute  to  produce  the 
type  of  current-voltage  characteristic  generally  obtained  in  prac- 
tical thermionic  tubes.  We  shall  therefore  assume  that  both 
cathode  and  anode  are  equipotential  parallel  surfaces  of  infinite 
extent,  and  that  the  electrons  emerge  from  the  hot  cathode  with 
zero  velocity.  The  cathode  C  and  anode  P  (Fig.  19)  will  be  sup- 
posed to  be  in  an  enclosure  in  which  a  perfect  vacuum  is  main- 
tained. The  degree  of  vacuum  necessary  to  approximate  this  per- 
fection will  be  discussed  in  the  next  chapter. 

The  cathode  C  can  be  raised  to  any  desired  temperature.  Let 
the  anode  P  be  raised  to  a  potential  Vi,  while  the  cathode  remains 
grounded.  As  long  as  the  temperature  of  C  is  so  low  that  practi- 
cally no  electrons  are  emitted,  the  potential  gradient  between  the 

1  J.  J.  THOMSON,  Conduction  of  Electricity  through  Gases,  2d  Ed.,  p.  223. 

2  C.  D.  CHILD,  Phys.  Rev.  Vol.  32,  p.  498,  1911. 

3 1.  LANGMUIR,  Phys.  Rev.,  (2),  Vol.  2,  p.  450,  1913. 

4  SCHOTTKY,  Jahrb.  d.  Radioaktivitat  u.  Elektronik,  Vol.  12,  p.  147,  1915, 


56 


THERMIONIC  VACUUM  TUBE 


plates  is  practically  constant  so  that  the  potential  distribution  is  a 
linear  function  of  X,  the  distance  from  the  plate  C,  and  can  be 
represented  by  a  straight  line  OP.  But  this  is  no  longer  the  case 
when  C  emits  electrons.  These  electrons  are  pulled  over  to  P  by 
the  applied  electric  field  and  their  presence  in  the  space  between 
C  and  P  modifies  the  potential  distribution.  In  Section  9  it  was 
shown  that  if  the  plates  C  and  P  are  infinitely  large  so  that  the 

P 
C 


Distance  from  Cotthoole 
FIG.  19. 

lines  of  force  are  straight  the  potential  V  at  any  point  distant  x 
from  C  is  given  by 

d2V 


where  p  is  the  volume  density  of  the  charge, 
function  of  x.    Thus 


Now  p  is  itself  a 


(2) 


The  relation  between  the  potential  V  and  the  distance  x  cannot  be 
obtained  unless  the  form  of  the  function  x  is  known.  We  know 
that  the  density  of  electrons  near  C  is  greater  than  near  P.  It 
can  therefore  be  seen  in  a  general  way  that  because  of  the  presence 


PHYSICS  OF  THE  THERMIONIC  VALVE  57 

of  the  electrons  the  potential  distribution  curve  will  be  some- 
what of  the  nature  shown  by  OAP  (Fig.  19).  The  potential  gradi- 
ent at  the  cathode  is  still  positive,  but  if  the  temperature  of  the 
cathode  be  now  raised  until  the  potential  difference  applied 
between  C  and  P  is  not  high  enough  to  draw  all  the  emitted 
electrons  away  from  Ci  and  if  it  is  assumed  that  the  electrons 
are  emitted  from  C  with  zero  velocity,  the  potential  distribution 
curve  takes  a  shape  somewhat  like  OBP,  which  has  a  horizontal 
tangent  at  0.  This  means  that  the  potential  gradient  at  0  is  zero. 
Any  further  increase  in  the  number  of  emitted  electrons  would 
tend  to  depress  the  curve  at  0  below  the  horizontal  line  OX. 
This  would  be  equivalent  to  establishing  a  negative  potential 
gradient  at  the  cathode,  which  would  tend  to  return  the  emitted 
electrons  to  the  cathode. 

The  assumption  that  the  electrons  emerge  from  the  cathode 
with  zero  velocity  does  not  lead  to  a  correct  description  of  the  true 
state  of  matters.  The  electrons  are  actually  emitted  with  finite 
velocities  which  are  distributed  according  to  Maxwell's  distribu- 
tion law.  This  law  forms  the  basis  of  Richardson's  equation. 
In  actual  practice  the  finite  velocities  with  which  the  electrons 
are  emitted  from  C  cause  the  potential  distribution  curve  to  be 
depressed  slightly  below  the  axis  OX  so  that  there  actually  exists 
at  the  cathode  a  slight  negative  potential  gradient.  This  will  be 
discussed  more  fully  below.  For  the  present  we  shall  assume 
that  the  electrons  start  from  the  cathode  with  zero  velocity.  The 
velocity  they  acquire  on  their  way  to  the  anode  P  is  then  due  en- 
tirely to  the  potential  difference  between  C  and  P.  The  kinetic 
energy  of  an  electron  at  a  point  distant  x  from  C,  i.e.,  where  the 
potential  is  V,  is  given  by 


(3) 


This  is  obtained  from  equation  (4)  Chapter  I,  by  putting  the  initial 
velocity  vo  =  0. 

It  was  shown  in  Section  4  that  if  p  is  the  number  of  electrons 
per  cubic  centimeter  multiplied  by  the  electronic  charge,  that  is, 
if  p  is  the  volume  density  of  electricity  or  space  charge,  the  motion 
of  the  electrons  constitutes  a  current  per  square  centimeter  given 
by: 

i=Pv.  ......    (4) 


58  THERMIONIC  VACUUM  TUBE 

where  v  is  the  velocity.  Both  p  and  v  are  functions  of  the  distance 
from  the  cathode  C,  but  the  product  pv  is  constant,  since  the  num- 
ber of  electrons  passing  per  second  through  unit  area  perpendicu- 
lar to  the  direction  of  the  electric  field,  that  is,  the  current  i  must 
be  the  same  for  all  points  between  cathode  and  anode. 

The  quantities  p  and  v  in  the  above  equations  (1),  (3)  and  (4), 
are  unknown  functions  of  x,  and  can  be  eliminated  from  these 
equations.  This  gives: 

dfF  =  2. 
The  integration  of  this  equation  gives 


where  VQ  is  the  potential  of  the  cathode  and  ( -7— • )   the  potential 

gradient  at  the  cathode.  Now,  the  potential  of  the  cathode  is 
supposed  to  be  zero,  and  since  the  initial  velocity  of  emission  of  the 
electrons  is  assumed  to  be  zero,  the  limiting  condition  for  full 

space  charge  is  that  ( -j- }  =  0.     Hence  (6)  becomes 

\ax  /o 


'-« 


Integrating  this  and  putting  as  limits  F  =  0  for  z  =  0  and  V  =  E 
for  x  =  d,  the  distance  between  the  plates,  we  obtain  the  equation 
of  the  current  as  a  function  of  E  and  d,  namely: 


(8) 


This  is  the  so-called  f  -power  equation  first  derived  by  Child,  which 
gives  the  relation  between  the  voltage  and  current  carried  by 
dislodged  charges  of  one  sign  only. 

By  putting  the  value  of  —  in  (8)  and  writing  A  for  the  area 
of  the  cathode  the  current  can  be  expressed  by 

......     (9) 


PHYSICS  OF  THE  THERMIONIC  VALVE  59 

where  i  is  the  current  in  amperes,  d  the  distance  between  the  plates 
in  centimeters  and  V  the  potential  difference  between  the  plates  in 
volts. 

It  will  be  noticed  that  the  space  charge  equation  gives  the  cur- 
rent in  terms  of  only  the  applied  voltage  and  the  geometry  of  the 
device.  The  space  charge  current  does  not  depend,  as  in  the  case 
of  the  saturation  current,  upon  the  temperature  or  electron  affinity 
of  the  cathode.  On  the  other  hand  the  saturation  current  is 
independent  of  the  distance  between  cathode  and  anode,  except 
that  if  this  distance  be  made  smaller  the  saturation  current  would 
be  obtained  at  a  lower  minimum  voltage.  Referring  to  the  curve 
OAiBi  (Fig.  17),  it  will  be  seen  that  the  space  charge  equation 
gives  the  current  for  values  of  the  voltage  less  than  that  corre- 
sponding to  A  i.  If  the  distance  between  cathode  and  anode  be 
decreased  the  curve  obtained  would  be  OHB\. 

26.  Quantitative  Relation  for  Concentric  Cylinders.  Ther- 
mionic tubes  are  frequently  made  in  the  form  in  which  the  anode 
is  a  cylinder  and  the  cathode  a  wire  stretched  along  its  axis.  The 
quantitative  relation  for  the  characteristic  of  such  a  device  was 
published  by  Langmuir  1  by  making  the  assumptions  that  both 
cathode  and  anode  are  infinitely  long,  and  both  are  equipotential 
surfaces.  Here,  also,  the  electrons  are  assumed  to  emerge  from 
the  cathode  with  zero  velocity.  In  the  next  section  it  will  be 
shown  what  modifications  must  be  made  in  order  to  make  the 
results  conform  mere  nearly  with  practical  conditions. 

The  equation  (3),  for  the  velocity  of  the  electrons,  can  be 
applied  directly  to  this  case.  The  differential  equation  for  the 
potential  distribution,  however,  now  takes  the  form 

d(  dV\ 

—  (  r— r-    =47rpr, (10) 

ar\  ar  / 

where  r  is  the  distance  from  the  cathode,  and  £he  equation  for  the 
current  becomes 

(ID 


where  i  is  the  current  per  unit  length. 

These  equations  and  equation  (3)  can  be  combined  to  give: 


d?V  .  dV     .    2m 


1 1.  LANGMUIR,  loc.  cit.,  p.  457. 


THERMIONIC  VACUUM  TUBE 


Putting  V  =  E,  the  potential  difference  between  cathode  aid 
anode,  when  r  is  the  radius  of  the  cathode,  the  solution  of  this 
equation  can  be  expressed  in  the  form: 


2    \2e 


(13) 


where  0  is  a  constant  which  is  determined  by  the  ratio  -  of  the  radii 

Cv 


of  the  anode  and  the  filament.     The  relation  between 
is  given  in  the  following  table  taken  from  Langmuir's  paper. 

TABLE  I 


and  - 
a 


r 

r 

a 

a 

1.00 

0.000 

5.0 

0.775 

1.25 

0.045 

6.0 

0.818 

1.50 

0.116 

7.0 

0.867 

1.75 

0.200 

8.0 

0.902 

2.00 

0.275 

9.0 

0.925 

2.50 

0.405 

10.0 

0.940 

3.00 

0.512 

15.0 

0.978 

4.00 

0.665 

00 

1.000 

It  is  seen  that  for  most  practical  cases  /3  can  be  put  equal  to 
unity.  If  this  is  done  and  the  length  of  the  filament  be  put 
equal  to  I,  equation  (13)  may  be  written: 


(13a) 


Substituting  the  value  of  —  and  reducing  i  to  amperes,  V  to  volts 

171 

and  the  radius  r  to  centimeters,  equation  (13a)  becomes 


14.65X10-6-- 
r 


(14) 


where  I  is  the  length  of  the  filament  in  centimeters. 

The  only  difference  between  equations  (9)  and  (14),  besides 
the  numerical  value  of  the  constant,  is  that  in  the  case  of  plane 


PHYSICS  OF  THE  THERMIONIC  VALVE 


parallel  electrodes  the  current  varies  inversely  as  the  square  of 
the  distance  d  between  the  electrodes,  whereas  in  the  case  of  the 
cylindrical  structure  the  current  varies  inversely  as  the  first 
power  of  the  radius  r  of  the  anode.  This  is  an  important  property 
when  considering  the  design  of  thermionic  tubes  with  very  small 
electrostatic  capacity,  such  as  is  required  when  operating  at 
extremely  high  frequencies. 

27.  Influence  of  Initial  Velocities.  The  two  main  assumptions 
underlying  the  derivation  of  equations  (8)  and  (13)  are  that  the 
electrons  emerge  from  the  cathode  with  zero  velocities  and  that 


ZO 


40 


60 


80  ICO 

Anode  Voltage 

FIG.  20. 


rco 


140 


160 


180 


the  cathode  is  an  equipotential  surface.  Let  us  first  see  how  the 
conditions  'are  altered  when  we  do  not  ignore  the  effect  of  the 
initial  velocities. 

By  assuming  zero  initial  velocities,  the  condition  is  obtained 
that  the  potential  gradient  at  the  cathode  is  zero.  This  means 
that  the  potential  distribution  curve  OBP,  shown  in  Fig.  19,  has 
a  horizontal  tangent  at  0.  The  resulting  equation  (8)  states  that 
the  current  varies  as  the  f-power  of  the  applied  voltage  for  all 
voltages  up  to  that  necessary  to  give  the  saturation  current. 
This  characteristic  is  shown  in  Fig.  20.  The  part  OAB  represents 
the  current  that  is  limited  only  by  the  space  charge  of  the  electrons 


62 


THERMIONIC  VACUUM  TUBE 


in  the  space  between  cathode  and  anode.  At  B  the  applied  voltage 
becomes  large  enough  to  pull  all  the  electrons  to  the  anode  as  fast 
as  they  are  emitted  from  the  cathode.  The  part  BC  represents 
the  saturation  current. 

The  effect  of  the  initial  velocities  can  be  understood  by  referring 
to  Fig.  21.    Suppose  the  anode  were  insulated  and  connected  to  a 


FIG.  21. 

pair  of  quadrants  of  an  electrometer.  If  the  cathode  be  raised 
to  a  high  temperature,  the  anode  would  acquire  a  negative  charge 
which  can  be  measured  with  the  electrometer.  Obviously  in  this 
case  the  potential  at  all  points  between  cathode  and  anode  must 
be  negative  and  the  potential  distribution  can  be  represented  by  a 
curve  such  as  the  curve  a  of  Fig.  21.  In  this  diagram,  as  in  Fig. 


PHYSICS  OF  THE  THERMIONIC  VALVE  63 

19,  the  abscissae  represent  distances  from  the  cathode,  and  the 
ordinates  the  potentials  with  respect  to  that  of  the  cathode  which 
we  shall  take  as  zero.  The  anode  will  charge  up  negatively  until 
the  potential  to  which  it  rises  is  so  high  that  the  field  at  all  points 
between  anode  and  cathode  is  sufficiently  strong  to  prevent  any 
more  electrons  from  reaching  the  anode  under  their  own  kinetic 
energy  of  emission.  If  the  anode  now  be  connected  to  the  cathode 
through  a  battery  which  maintains  the  anode  at  a  small  positive 
potential,  the  potential  distribution  can  be  represented  by  the 
curve  b  of  Fig.  21.  This  curve  shows  that  the  potential  gradient 
is  still  negative  at  the  cathode,  but  in  passing  from  cathode  to 
anode  the  gradient  passes  through  zero  at  a  distance  x  from  the 
cathode  and  then  becomes  positive.  The  negative  gradient  at 
distances  less  than  x  is  due  to  the  space  charge  of  the  electrons  and 
the  fact  that  the  energy  of  emission  enables  the  electrons  to  move 
against  a  negative  potential  gradient.  If  the  anode  potential  be 
increased  the  distance  x  of  minimum  potential  shortens  until  the 
potential  distribution  finally  takes  the  shape  given  by  curve  d. 
In  this  case  x  is  practically  zero,  so  that  the  potential  gradient 
vanishes  at  the  cathode.  When  this  condition  is  reached  we  have 
the  case  corresponding  to  the  point  B  of  Fig.  20.  If  the  plate 
potential  be  increased  still  further  the  potential  distribution 
straightens  out  still  more  (curve  e)  and  the  current  obtained  is  the 
saturation  current. 

Now,  it  is  the  condition  represented  by  the  potential  distribu- 
tion curve  d  that  was  assumed  in  the  derivation  of  the  f-power 
equation.  But  this  is  the  condition  obtaining  when  the  space 
current  just  becomes  equal  to  the  saturation  current.  It  is 
therefore  to  be  expected  that  on  account  of  the  initial  velocities, 
Child's  f-power  equation  would  hold,  strictly  speaking,  only  for 
currents  approximating  the  saturation  current. 

The  explanation  given  here  for  the  influence  of  the  initial 
velocities  of  the  emitted  electrons  is  based  on  the  solution  of  the 
problem  furnished  me  by  T.  C.  Fry,  who  also  computed  the  dis- 
tance x  of  the  minimum  potential  for  cases  approximating  the 
conditions  met  with  in  practice.  The  values  of  x  depend  on  the 
distance  between  the  electrodes  and  the  potentials  applied  to  the 
anode.1 

1  The  effect  of  the  initial  velocities  has  also  been  studied  by  W.  SCHOTTKT 
(Phys.  Zeitschr.,  Vol.  15,  1914),  P.  S.  EPSTEIN  (Deutsch  Phys.  Gesell.  Verh. 
21,  p.  85-99,  1919)  and  others. 


64 


THERMIONIC  VACUUM  TUBE 


The  effect  of  the  initial  velocities  on  the  shape  of  the  current- 
voltage  characteristic  is  demonstrated  in  Fig.  22,  where  the 
logarithms  of  the  currents  are  plotted  against  the  logarithms 
of  the  potential  differences  applied  between  anode  and  cathode. 
The  characteristic  shown  in  Fig.  20  gives  on  the  logarithmic  plot 
a  straight  line  AB  having  a  slope  equal  to  f .  On  account  of  the 
initial  velocities,  the  logarithmic  plot  takes  the  form  given  by  the 
line  CD,  the  slope  of  which  approximates  f  at  the  upper  part. 

28.  Effect  of  Voltage  Drop  in  the  Filament.  In  practice  the 
cathode  is  never  an  equipotential  surface,  but  takes  the  form  of  a 


Log  (Anode  Volts) 
FIG.  22. 


filament  which  is  rendered  incandescent  by  passing  a  current 
through  it.  There  is  consequently  established  a  voltage  drop  in 
the  filament  which  causes  a  marked  deviation  from  the  f-power 
equation.  This  deviation  is  in  an  opposite  sense  to  that  due  to  the 
initial  velocities. 

The  equation  for  the  characteristic  resulting  from  a  considera- 
tion of  the  effect  of  the  voltage  drop  in  the  filament  was  given  by 
W.  Wilson.1  This  derivation  is  also  based  on  the  assumption  that 
the  initial  velocities  can  be  neglected. 

1  Paper  read  at  the  Philadelphia  meeting  of  the  American  Physical  Society, 
Dec.,  1914. 


PHYSICS  OF  THE  THERMIONIC  VALVE  65 

Let  us  consider  a  structure  in  which  the  filament  is  a  single 
wire  stretched  along  the  axis  of  a  cylindrical  anode  and  let  the  nega- 
tive terminal  of  the  filament  be  grounded  so  that  we  may  regard 
its  potential  as  zero.  Let  the  potential  drop  in  the  filament,  due 
to  the  heating  current,  be  Ef  and  let  the  potential  of  an  interme- 
diate point  at  a  distance  x  from  the  negative  end  be  V.  If  I  be 
the  length  of  the  filament  we  have  for  constant  current  in  it  : 


If  E  be  the  potential  difference  between  the  anode  and  the  negative 
end  of  the  filament,  the  potential  difference  between  the  anode  and 
a  point  of  the  filament  which  is  at  a  potential  V  is 


and  this  is  zero  when 

x=E±. (16) 

This  means  that  current  will  only  flow  from  the  length  x  of  the 
filament  given  by  (16)  since  all  points  of  the  filament  beyond  x 
are  positive  with  respect  to  the  anode. 

Since  a  very  small  length  dx  of  the  filament  can  be  regarded 
as  an  equipotential  surface,  the  current  can  be  taken  to  vary  as 
the  f-power  of  the  potential  difference  between  the  anode  and  the 
point  x,  if  we  neglect  the  factors  such  as  the  initial  velocities, 
which  cause  a  deviation  from  the  simple  f-power  equation.  The 
potential  difference  between  the  anode  and  the  point  x  of  the  fila- 

/v» 

ment  is  E  —Et-.     Hence  if  Az  be  the  current  from  the  element 

I 

dx  of  the  filament,  we  get  by  equation  (13a) 

3/2 


In  order  to  obtain  the  total  current,  we  have  to  integrate  over 
the  length  of  the  filament  from  which  the  electrons  flow  to  the 
anode.  We  have  to  distinguish  two  cases:  (a)  When  the  voltage 
E  between  the  anode  and  the  negative  end  of  the  filament  is  less 
than  the  voltage  drop  Ef  in  the  filament,  due  to  the  heating  cur- 
rent, and  (6)  when  E  is  greater  than  Ef. 


66  THERMIONIC  VACUUM  TUBE 

Case  (a)  E^.Ef.     Here  the  integration  is  to  be  performed  over 
the  part  x  of  the  filament  given  by  equation  (16).     Thus 

-  -  i  2 

9r 
This  gives 

1*  •  ......  <"> 


When  #  is  expressed  in  volts  and  i  in  amperes,  we  may  write 
this  equation: 


Comparing  this  constant  of  proportionality  K  with  C  in  equation 
(14)  it  will  be  seen  that 

.     (18o) 


Equation  (18)  shows  that  as  long  as  the  potential  difference 
between  the  anode  and  the  negative  end  of  the  filament  is  less 
than  the  voltage  drop  in  the  filament,  the  anode  current  varies 
as  the  f  -power  of  the  anode  potential.  Except  for  the  fact  that 
here  the  limitation  of  current  by  the  voltage  drop  in  the  filament 
has  been  taken  care  of,  this  equation  is  subject  to  the  same  limita- 
tions as  the  f  -power  equations  which  were  derived  on  the  assump- 
tion that  the  cathode  is  an  equipotential  surface. 

Case  (b)  E^Ef.     In  this  case  electrons   flow  from   the  whole 
surface  of  the  filament  to  the  anode.     Hence  the  current  is: 


9r  \rnjo 
which  gives: 

(i)^Ef=K[^-(E-Ef)^],      .    .    ...    (19) 
where 


1  =  length  of  the  filament; 
r  =  radius  of  the  cylindrical  anode; 
Ef= voltage  drop  in  the  filament. 


PHYSICS  OF  THE  THERMIONIC  VALVE  67 

Equation  (19)  may  be  expanded  into  the  more  convenient  form: 


where  C=14.65X10~6-,  the  same  constant  as  appears  in  the 

equation  (14). 

The  lower  signs  in  the  series  of  (19a)  pertain  to  the  case  in  which 
the  potential  of  the  anode  is  reckoned  with  respect  to  the  positive 
instead  of  the  negative  end  of  the  filament. 


F1 


FIG.  23. 

This  series  converges  so  rapidly  that  for  all  values  of  the  anode 
potential  greater  than  twice  the  voltage  drop  in  the  filament  we  can 
write  for  the  current  with  close  approximation. 


Hf] 


(20) 


In  deriving  these  equations,  the  length  of  the  filament  (and  that 
of  the  cylindrical  anode)  were  put  equal  to  a  finite  value  I.  Strictly 
speaking  I  should  be  infinitely  long  so  that  the  distortion  of  the 
field  at  the  ends  of  the  anode  can  be  neglected.  This  condition 
can  be  realized  in  practice  with  a  device  shown  schematically  in 
Fig.  23. 

The  anode  AA  is  in  the  form  of  a  cylinder  and  the  filament  is 
stretched  along  its  axis.  In  order  to  insure  straight  lines  of 
force  the  guard  rings  RRr  are  placed  on  either  side  of  AA,  the 
filament  extending  beyond  the  ends  of  the  anode.  The  anode 
and  guard  rings  are  electrically  connected  but  the  galvanometer  G 


68  THERMIONIC  VACUUM  TUBE 

is  inserted  as  indicated  in  the  diagram,  so  that  it  registers  only  the 
electron  current  flowing  to  the  anode.  The  effective  length  of  the 
filament  is  then  equal  to  the  length  of  the  anode. 

The  general  effect  of  the  voltage  drop  in  the  filament  when  the 
plate  is  connected  to  the  negative  end  of  the  filament,  is  to  make 
the  space  current  smaller  because  the  average  potential  difference 
between  the  filament  and  the  plate  is  smaller  than  that  between 
the  negative  end  of  the  filament  and  the  plate,  this  being  the 
potential  difference  that  is  ordinarily  measured  with  a  voltmeter. 
The  general  effect  is  shown  by  the  curve  OD  of  Fig.  20.  The  curve 
OAB  represents  the  theoretical  curve  in  accordance  with  the  simple 
f -power  equation,  and  the  part  AB  represents  the  ideal  saturation 
current  which  is  supposed  to  be  independent  of  the  applied 
voltage.  The  characteristic  which  is  ordinarily  observed  is  indi- 
cated by  ODE.  For  the  present  we  shall  consider  only  the  lower 
part  OD  of  this  characteristic.  The  deviation  at  voltages  greater 
than  the  voltage  corresponding  to  the  point  D  is  due  to  the  limita- 
tion of  the  current  by  the  electron  emission  from  the  filament  and 
will  be  discussed  in  the  next  section.  The  line  OAB  is  computed 
from  the  f -power  equation  (14),  the  constant  C  being  put  equal 
to  50X10"6  amperes.  If  we  assume  that  the  voltage  drop  in  the 
filament  is  10  volts,  then  referring  to  equation  (18a),  we  find  the 
constant  K  becomes  equal  to  2  X  10~6.  With  the  help  of  equations 
(18)  and  (19)  we  can  then  compute  the  current  as  a  function  of  the 
potential  differences  E  between  filament  and  plate,  by  putting 
#/=10.  The  values  so  computed  give  the  value  OD  of  Fig.  20. 
The  percentage  deviation  of  this  curve  from  the  theroretical 
curve  OA  is  quite  considerable  at  the  lower  voltages.  It  will  be 
explained  in  the  next  chapter  that  it  is  desirable  to  so  design 
thermionic  tubes  that  the  saturation  current  is  obtained  at  the 
smallest  possible  voltage.  In  practice  the  voltage  necessary 
for  saturation  seldom  exceeds  a  few  hundred  volts.  Fig.  24 
represents  two  experimental  curves  plotted  on  the  logarithmic 
scale  and  obtained  in  such  a  way  that  in  the  one  case  (curve  2) 
the  voltage  drop  in  the  filament  was  effective  and  in  the  other 
(curve  I)  it  was  eliminated.  To  eliminate  the  effect  of  the 
voltage  drop  in  the  filament  we  can,  as  has  been  done  in  taking 
these  curves,  resort  to  a  scheme  used  by  von  Baeyer  1  in  1909, 
which  consists  in  connecting  the  ends  of  the  filament  and  the 
1  0.  VON  BAEYER,  Phys.  Zeits.,  Vol.  10,  p.  168,  1909. 


PHYSICS  OF'  THE  THERMIONIC  VALVE 


69 


plate  through  a  commutator  which  is  so  arranged  that  the  filament 
current  and  the  plate  voltage  are  applied  alternatively  for  short 
intervals  of  time,  the  plate  voltage  being  applied  only  while  the 
filament  voltage  is  cut  off.  If  the  alternations  are  frequent 
enough,  the  filament  does  not  get  a  chance  to  cool  off  markedly 
during  the  time  that  the  filament  current  is  cut  off.  In  this  way 
the  plate  current  is  measured  only  while  there  is  no  voltage  drop 
in  the  filament. 


Iliampere* 

ro  oJ  •*> 

-I  €30  tD  CD  O  .0  C 

/ 

o 

c 

/ 

o    ° 

0 

0 

> 

I 

c 

c 

9 

/ 

/ 

i 

& 

J 

, 

1 

> 

» 

2:  ' 

<u   6 
c    5 

4 
3 

y 

/ 

2 

^ 

2 

/ 

// 

7 

I 

/x 

/ 

X 

/ 

/ 

/ 

/ 

* 

5     6    ' 

3  9  10                   ZO          30      40     50  6 
Anode  Volts 

7   80° 

flIOO                 20 

FIG.  24. 

The  dotted  line  in  Fig.  24  is  drawn  to  have  a  slope  equal  to  f . 
For  the  higher  voltages  curve  1  is  probably  close  to  the  theoretical 
line,  but  at  the  lower  voltages  it  bends  to  the  left  on  account  of  the 
initial  velocities  of  the  emitted  electrons.  Curve  2  plainly  shows 
the  effect  of  the  voltage  drop  in  the  filament;  it  deviates  every- 
where to  the  right  of  the  theoretical  line.  Hence,  the  effect  of 
the  voltage  drop  in  the  filament  is  opposite  to  the  effect  caused  by 
the  initial  velocities.  While  the  latter  causes  the  logarithmic 


70  THERMIONIC  VACUUM  TUBE 

plot  of  the  characteristic  to  deviate  to  the  left  from  the  f-line,  the 
voltage  drop  in  the  filament  causes  not  only  a  deviation  to  the  right 
but  also  a  lateral  shift  of  the  whole  curve  to  the  right.  These 
two  effects  sometimes  contribute  to  give  a  better  f  logarithmic 
line  than  is  to  be  expected  from  theoretical  considerations  only, 
because  their  effects  are  opposite  and  tend  to  neutralize  each  other. 
This  is  especially  the  case  when  the  filament  is  operated  at  a  very 
high  temperature,  because  the  higher  the  temperature  of  the 
filament,  the  greater  will,  of  course,  be  the  effect  of  the  initial 
velocities. 

As  far  as  the  lower  part  of  the  characteristics  of  thermionic 
valves  is  concerned,  it  follows,  therefore,  that  the  current  is  limited 
not  only  by  space  charge  but  by  the  initial  velocities  of  emission  of 
electrons  and  by  the  voltage  drop  in  the  filament  as  well.  If  we 
consider  the  whole  characteristic  over  which  it  is  sometimes 
operated,  we  find  that  the  current  is  also  limited  by  another  fac- 
tor which  we  will  now  proceed  to  explain. 

29.  Influence  of  Limitation  of  Current  by  Thermionic  Emission. 
If  we  consider  the  characteristics  as  actually  obtained  in  practice, 
we  find  that  on  the  upper  parts  they  deviate  even  more  from  the 
theoretical  line  than  the  lower  part.  Referring,  for  example,  to  Fig. 
20,  the  curve  ODE  represents  more  nearly  what  is  actually  obtained 
with  thermionic  tubes,  which  is  quite  different  from  the  theoretical 
curve  OAB.  As  we  proceed  up  the  characteristic  from  the  lower 
voltage  values,  we  find  that  the  curve  deviates  from  the  theoretical 
curve  on  account  of  the  voltage  drop  in  the  filament,  but  as  we 
proceed  higher  up  the  characteristic  the  curve  bends  over  gradually 
toward  saturation.  The  two  things  to  be  noticed  are  that  this 
transition  region  between  the  lower  part  and  the  saturation  part 
of  the  characteristic  generally  starts  at  comparatively  low  voltages 
and,  secondly,  the  saturation  current  itself  usually  increases  gradu- 
ally up  to  very  high  voltages  instead  of  becoming  flat,  as  indicated 
by  AB.  This  deviation,  which  is  shown  by  the  part  DE,  Fig.  20, 
is  due  to  the  fact  that  in  the  neighborhood  of  D  the  voltage 
approaches  values  at  which  the  current  begins  to  be  limited  by 
electron  emission.  The  gradual  bending  of  the  curve  and  the 
length  of  this  transition  region  depend  on  the  surface  conditions 
of  the  cathode  the  voltage  drop  in  the  filament  and  on  the  shape  of 
the  anode.  With  the  Wehnelt  cathode  the  transition  part  of  the 
curve  is  generally  much  longer  than  with  Tungsten  cathode. 


PHYSICS  OF  THE  THERMIONIC  VALVE  71 

Furthermore,  if  the  anode  is  in  the  form  of  a  wire  or  a  small  plate, 
the  transition  part  is  also  longer  than  when  the  anode  is,  for  ex- 
ample, a  cylinder  surrounding  the  cathode  placed  along  its  axis. 

The  transition  part  of  the  characteristic  can  best  be  seen  from 
diagrams  in  which  are  plotted  a  number  of  characteristics  obtained 
with  the  same  tube  but  with  different  filament  temperatures 
such  as  shown,  for  example,  in  Fig.  25.  Taking,  for  example,  the 
lowest,  one  of  these  curves,  we  see  that  the  transition  part  sets  in 
at  A.  In  the  second  curve,  it  sets  in  at  5,  and  so  on. 

Generally  speaking,  practical  thermionic  devices  operate  over 
the  part  of  the  characteristic  indicated  by  ODE  of  Fig.  20.  This 
part  we  can  refer  to  as  the  infra-saturation  part  or  the  operating 
part  of  the  characteristic.  From  the  above  explanations  it  follows 
that  over  this  operating  part  of  the  characteristic  the  current  is 
limited  by  space  charge,  initial  velocities  of  emission,  the  voltage 
drop  in  the  filament  and  by  thermionic  emission.  There  is  another 
factor  which  limits  the  current  on  the  lower  voltages,  namely, 
the  formation  of  heavy  negative  carriers  when  a  small  amount 
of  gas  is  present  in  the  tube.  This  will  be  discussed  more  fully 
in  the  next  chapter.  In  the  case  of  the  three-electrode  tube 
which  contains  a  grid  inserted  between  the  filament  and  the  plate, 
the  current  is  further  limited  by  the  grid  (see  Chapter  VII).  All 
these  current  limiting  factors  contribute  to  the  production  of  a 
characteristic  which  is  curved  and  which  does  not  obey  a  simple 
power  law  when  taken  over  the  whole  infra-saturation  or  operating 
range  of  the  characteristic.  The  logarithmic  plot  of  the  character- 
istic is  steepest  at  the  lower  voltages  where  the  slope  may  be  as 
high  as  2J.  As  the  voltage  increases  the  slope  of  the  logarithmic 
plot  decreases  until  finally  it  becomes  less  than  unity  when  satura- 
tion is  approached. 

When  dealing  with  certain  small  parts  of  the  characteristic, 
we  can  advantageously  apply  a  simple  power  law.  Thus,  in  the 
case  of  a  tube  containing  a  grid,  we  operate  the  tube  as  an  amplifier 
generally  over  a  range  on  the  lower  part  of  the  characteristic  where 
the  relation  between  current  and  voltage  can  be  expressed  as  a 
simple  quadratic  relation  (see  Chapter  VII).  When  using  the 
three-electrode  tube  as  an  oscillation-generator,  on  the  other  hand, 
we  use  the  whole  characteristic  ranging  all  the  way  up  to  saturation 
voltages. 

In  order  to  obtain  a  rough  indication  of  the  maximum  current 


72 


THERMIONIC  VACUUM  TUBE 


that  can  be  passed  through  the  tube,  we  can  apply  the  simple 
f-power  equation  in  special  cases.  Thus,  if  the  anode  is  a  cylinder 
and  the  cathode  stretched  along  its  axis,  the  application  of  equation 


80  120  160 

Anode  Volts 

FIG.  25. 


280 


(14)  will  give  an  approximate  indication  of  the  maximum  current 
that  can  be  passed  through  the  tube  at  voltages  approximating 
to  the  saturation  voltage. 


PHYSICS  OF  THE  THERMIONIC  VALVE  73 

The  application  of  the  f-power  equation  to  the  approximate 
determination  of  the  current  at  the  saturation  voltage,  is  not  based 
on  the  consideration  in  Section  27  where  it  was  shown  that  theo- 
retically the  f-power  equation  holds  for  an  equipotential  cathode 
at  voltages  approximating  the  saturation  voltage,  because  the 
limitation  of  current  by  electron  emission  which  makes  itself  felt 
even  on  the  lower  part  of  the  characteristic,  causes  quite  a  marked 
deviation  from  the  theoretical  curve,  so  that  the  f-power  equa- 
tion can  be  used  only  to  give  a  very  rough  indication  of  the 
maximum  current. 

It  is  not  to  be  concluded  that  since  the  initial  velocities  and 
the  voltage  drop  in  the  filament  both  exert  effects  which  become 
proportionately  smaller  as  the  voltage  increases,  an  agreement  with 
the  f-power  law  can  be  demonstrated  if  the  upper  part  of  the  char- 
acteristic gives  a  slope  equal  to  f .  The  fact  of  the  matter  is  that 
on  account  of  the  voltage  drop  in  the  filament  the  slope  of  the 
characteristic  well  below  the  saturation  part  is  greater  than  f 
and  on  the  saturation  part  itself  it  is  less  than  unity  and  sometimes 
almost  zero.  Since  the  slope  changes  actually  from  the  high 
value  to  the  low  value,  there  must,  of  course,  be  a  region  where 
the  characteristic  has  a  slope  equal  to  f .  But,  in  this  region 
the  current  is  limited  not  only  by  space  charge  but  by  the  voltage 
drop  in  the  filament  and  by  thermionic  emission  from  the  filament 
as  well.  This  effect  can  readily  be  seen  by  plotting  the  curves 
shown  in  Fig.  25  on  the  logarithmic  scale,  as  is  shown  in  Fig.  26. 
It  will  be  seen  that  each  of  these  curves  has  a  slope  equal  to  f 
over  a  more  or  less  restricted  region.  The  three  different  curves 
give  three  different  lines  each  having  a  slope  equal  to  f .  This 
would  mean  that  this  one  tube  obeys  three  different  f-power 
equations  which  is,  of  course,  impossible.  For  the  f-power 
equation  to  be  obeyed,  it  is  not  only  necessary  that  the  logarith- 
mic plot  give  a  line,  the  slope  of  which  is  f ,  but  that  line  must,  of 
course,  also  have  a  definite  intercept  on  the  voltage  axis  which 
gives  the  constant  of  proportionality  of  the  f-power  equation. 

30.  Effect  of  the  Curvature  of  the  Characteristic.  In  the  pre- 
vious section  it  was  explained  how  a  number  of  factors  contribute 
to  limiting  the  current  in  such  a  way  as  to  produce  a  characteristic 
which  is  curved.  Of  these  factors  the  space  charge  of  the  electrons 
in  the  space  between  filament  and  anode  is  responsible  for  the  fact 
that  the  characterictic  curves  upwards.  The  limitation  of  current 


74 


THERMIONIC  VACUUM  TUBE 


by  electron  emission,  on  the  other  hand,  causes  the  characteristic 
gradually  to  bend  over  to  the  right.  The  fact  that  the  characteristic 
is  curved  is  a  very  great  disadvantage  in  a  large  number  of  uses 
to  which  the  tube  is  applied.  As  will  be  explained  later,  it  causes 
the  tube  to  distort  when  it  is  used  as  an  amplifier.  When  the  tube 
is  used  as  an  oscillation-generator,  the  curvature  of  the  character- 


10 

£8 


40  60 

Anode  Volts 


FIG.  26. 


200 


tic  is  also  a  disadvantage  in  that  it  makes  the  solution  of  the 
problem  of  the  oscillator  so  much  more  difficult.  In  fact,  the  full 
solution  for  the  curved  characteristic  has  not  yet  been  given. 
On  the  other  hand,  when  the  tube  is  used  as  a  modulator  or  detector 
of  high  frequency  oscillations  the  curvature  of  the  characteristic 
is  actually  made  use  of.  In  considering  the  applications  of  the 
thermionic  tube,  we  shall  first  deal  with  those  for  which  it  is  desir- 


PHYSICS  OF  THE  THERMIONIC  VALVE  75 

able  to  have  a  straight  characteristic  and  then  consider  the  applica- 
tions which  make  use  of  the  curvature  of  the  characteristic. 

Whatever  has  been  explained  here  with  reference  to  the  simple 
two-electrode  valve  applies  in  a  general  way  and  with  certain 
qualifications  also  to  the  much  more  important  type  of  thermionic 
device,  the  three-electrode  tube  which  contains  a  grid  between  the 
filament  and  the  plate. 

31.  Energy  Dissipation  at  the  Anode.  When  a  thermionic 
tube  is  in  operation  its  anode  becomes  heated  to  an  extent  de- 
pending on  the  size  and  nature  of  its  surface  and  on  the  voltage 
and  current  at  which  the  tube  is  operated.  If  v  be  the  velocity 
with  which  an  electron  strikes  the  anode,  the  energy  converted  into 
heat  at  the  anode  is  \mv1.  If  the  number  of  electrons  striking  the 
anode  per  second  is  n  the  power  dissipated  is 


where  I  is  the  current  through  the  tube  and  E  the  voltage  between 
anode  and  cathode.  The  temperature  of  the  anode  will  rise  until 
the  rate  at  which  energy  is  transferred  to  it  by  the  bombarding 
electrons  becomes  equal  to  the  rate  at  which  it  is  radiated  from  it. 
If  A  be  the  area  of  the  anode  and  e  its  thermal  emissivity,  the 
energy  radiated  per  second  can  be  given  approximately  by  the 
Stef  an-Boltzmann  equation  : 

P  =  keA(T4-T04),    .     .     .^.     .     (21) 

where  T  is  the  temperature  of  the  anode  in  Kelvin  degrees  and 
TQ  that  of  its  surroundings.  The  radiation  constant  k  is  the  power 
radiated  by  1  cm.2  of  a  perfect  black  body  at  a  temperature  of 
1°  K,  its  surroundings  being  at  absolute  zero  of  temperature. 

For  tungsten  Langmuir  l  finds  the  following  equation  instead 
of  the  above  equation  (21): 

/     fi    \4.74 

P=  12.54^4)     ......     (22) 

When  equilibrium  is  established  the  power  radiated  is  equal 

to  the  kinetic  energy  of  the  bombarding  electrons  converted  per 

second  into  heat.     The  term  To4  in  equation  (21)  can  generally 

be  neglected,  when  the  anode  temperature  is  high  and  we  obtain 

1  1.  LANGMUIR,  Phys.  Rev.,  Vol.  34,  p.  401,  1912. 


76  THERMIONIC  VACUUM  TUBE 

the  following  relation  between  the  power  dissipation  at  the  anode 
and  the  temperature  to  which  it  rises: 

EI  =  keAT4 (23) 

The  constant  €  has  its  maximum  value  unity  when  the  surface 
has  the  full  radiating  power  of  a  black  body.  A  shiny  surface  does 
not  radiate  as  well  as  a  dull  black  or  dark  surface,  and  is  therefore 
not  as  suitable  for  use  in  high  power  tubes.  Furthermore,  the 
power  radiated  depends  upon  the  area  of  the  anode. 

These  factors  have  to  be  considered  in  the  design  of  power 
tubes.  The  area  of  the  anode  must  be  so  proportioned  with  respect 
to  the  product  El  that  its  temperature  does  not  rise  beyond  a 
certain  value  depending  upon  the  nature  of  the  anode  and  the 
extent  to  which  it  has  been  denuded  of  gas  during  evacuation  of 
the  tube. 

If  the  anode  temperature  becomes  too  high  three  deleterious 
effects  can  come  into  play: 

(a)  It  can  cause  the  liberation  of  gas  and  so  make  the  discharge 
depart  from  a  pure  electron  discharge.  This  impairs  the  operation 
of  the  tube  and  often,  when  it  does  happen,  causes  the  tube  to  blow 
out  on  account  of  the  increase  in  current  due  to  ionization  and  the 
consequent  increase  in  power  dissipation  at  the  anode.  Such 
impairment  is  often  only  transient  since  the  gas  is  usually  cleaned 
up  by  the  hot  filament.  (See  Chapter  V.) 

(6)  It  causes  a  volatilization  of  the  anode  resulting  in  the 
formation  of  a  metallic  deposit  on  the  bulb  ("  blackening  of  the 
bulb  ").  Unless  the  parts  where  the  lead-in  wires  are  sealed  in 
are  shielded  the  metallic  deposit  impairs  the  insulation  between 
the  wires  which  must  in  most  cases  be  very  good. 

(c)  It  can  cause  the  emission  of  electrons  from  the  anode. 
This  is  generally  not  so  serious  in  the  case  of  a  three-electrode 
tube,  consisting  of  filament,  grid  and  anode  because  there  usually 
exists  a  strong  electric  field  between  the  grid  and  the  anode  which 
tends  to  return  to  the  anode  all  electrons  emitted  from  it.  But 
in  the  case  of  a  valve,  that  is,  a  tube  containing  only  cathode  and 
anode,  care  must  be  taken  that  emission  of  electrons  from  the 
anode  does  not  take  place,  because  if  it  did,  the  tube  would  not 
rectify  completely. 

32.  Efficiency  of  the  Cathode.  The  efficiency  of  a  thermionic 
cathode  is  determined  by  two  factors:  its  life,  and  the  maximum 


PHYSICS  OF  THE  THERMIONIC  VALVE 


77 


thermionic  current  obtainable  for  a  given  amount  of  power  ex- 
pended in  maintaining  it  at  the  desired  temperature.  The  satura- 
tion current,  i.e.,  the  maximum  obtainable  current,  depends  upon 
the  area  of  the  cathode,  its  temperature  and  "its  electron  affinity. 
(Equation  7,  Chapter  III.)  The  power  necessary  to  maintain  the 
cathode  at  a  definite  temperature  depends  upon  its  area,  its  tem- 
perature and  its  thermal  emissivity  (equation  23) .  By  comparing 
these  two  equations  it  will  be  seen  that  the  saturation  current  in- 
creases more  rapidly  with  the  temperature  than  does  the  power. 
Hence  the  saturation  current  per  unit  power  increases  with  the 
temperature.  The  relation  between  these  quantities  is  shown  in 
the  following  table  which  gives  the  values  for  tungsten  compiled 
from  a  paper  by  Dushman.1  The  second  column  gives  the  power 
in  watts  per  cm.2  necessary  to  maintain  the  tungsten  cathode  at 
the  temperature  given  in  the  first  column,  and  the  third  column 
gives  the  saturation  thermionic  current  in  milliamperes  per  cm.2 
of  cathode  surface.  The  numbers  given  under  s  in  the  last  column 
are  obtained  by  dividing  Is  by  p  and  can  be  taken  as  the  thermionic 
efficiency  of  tungsten  expressed  in  milliamperes  per  watt, 


TABLE  II 


T 

Kelvin  Degrees. 

P 
Watts  per  Cm2. 

Is 

Mils  per  Cm.2. 

s 
Mils  per  Watt. 

1000 

0.9 

1.2X10-11 

1.  25X10-" 

1500 

6.9 

6X10~4 

8.7X10-5 

1800 

16.4 

3X10-1 

1.8X10-2 

2000 

26.9 

4.2 

1.6X10-1 

2100 

34* 

15.1 

4.5X10"1 

2200 

43* 

48.3 

1.12 

2300 

53* 

137.7. 

2.6 

2400 

65* 

364.8 

5.6 

2500 

77.5 

891.0 

11.5 

2600 

90* 

2044 

22.7 

*  Obtained  by  interpolation. 

For  temperatures  above  1800°  K.  the  relation  between  the 
thermionic  efficiency  s  and  the  power  p  per  cm.2  expended  in 

1  S.  DUSHMAN,  G.  E.  Rev.,  Vol.  18,  156,  1915. 


78  THERMIONIC  VACUUM  TUBE 

heating  the  filament  can  be  expressed  with  a  sufficient  degree  of 
accuracy  by  the  simple  equation 

s  =  Cpn    .    .    .     .....     (24) 

where  C  and  n  are  constants. 

From  this  it  follows  also  that  for  the  operating  range  of  tem- 
peratures above  1800°  K.,  we  can  instead  of  equation  (7),  Chapter 
III,  use  the  simpler  equation 


(25) 


For  tungsten  the  constants  n  and  C  have  the  values,  n  =  4.13 
and  C=  1.812  X10~7  when  s  is  expressed  in  milliamperes  per 
watt,  p  in  watts  per  cm.2  and  i  in  milliamperes  per  cm.2 

The  important  quantity  is  s.  In  designing  tubes  it  is  not 
always  necessary  to  know  the  temperature  of  the  cathode.  Instead 
of  the  temperature  we  can  use  the  quantity  p,  because,  after  all, 
we  are  interested  only  in  the  power  that  must  be  expended  in  heat- 
ing the  filament  to  obtain  the  desired  thermionic  current.  This 
power  should  not  be  made  too  high  as  this  would  decrease  the  life 
of  the  filament.  If  the  desired  life  is  known,  we  can,  from  a  relation 
between  the  life  and  the  power  p  dissipated  per  cm.2  of  the  fila- 
ment, determine  the  value  of  p  at  which  the  filament  must  be 
operated.  From  the  power  that  the  tube  is  required  to  give  in 
its  plate  circuit  and  the  permissible  voltage  between  filament  and 
plate  the  saturation  current  can  be  determined.  With  the  help 
of  a  table  such  as  the  above  the  required  area  cf  the  filament  can 
then  be  obtained.  Once  this  area  is  known  the  length  and  diam- 
eter of  the  filament  can  be  determined  from  its  resistivity  at  the 
operating  temperature.  The  length  and  diameter  can,  of  course, 
be  proportional  to  suit  the  voltage  or  current  at  which  it  is  desired 
to  operate  the  filament. 

The  most  important  of  the  factor  that  influences  the  thermionic 
efficiency  of  a  filament  is  its  electron  affinity,  i.e.,  the  work  which 
an  electron  must  do  to  escape  from  the  filament.  A  glance  at 
the  following  table  will  show  the  enormous  extent  to  which  this 
constant  can  influence  the  saturation  current.  This  table  also 
gives  an  indication  of  the  manner  in  which  the  saturation  current 
is  increased  by  increasing  the  temperature  of  the  filament.  The 


PHYSICS  OF  THE  THERMIONIC  VALVE 


79 


figures  given  in  the  table  are  only  relative,  and  were  computed 
from  the  equation 


.4343  X105 
—  -- 

o.O 


,   .     .     .     (26) 


where  A  was  put  equal  to  107.  This  is  the  logarithm  equivalent 
of  Richardson's  equation  of  thermionic  emission.  (See  equations 
(9)  and  (10),  Chapter  III.) 


TABLE  III 


Temperature 
of  Cathode. 


Saturation  Current  in  Amperes  per  Cm.2  of  Cathode 
Surface. 


0=2  Volts. 

0  =  3  Volts. 

0=4  Volts. 

0  =  5  volts. 

1000 

25X10-3 

2X10~7 

2X10~12 

2X10~17 

1500 

72 

3X10"2 

13X10"6 

6X10-9 

2000 

4X103 

12 

36X10~3 

i.ixio-4 

2500 

4.6X104 

43 

4.2 

4X10-2 

If  we  take  two  filaments  having  different  values  of  electron 
affinity,  but  both  filaments  having  the  same  area  and  thermal 
emissivity,  place  them  in  identical  tubes  and  dissipate  the  same 
power  in  them,  they  would  give  the  current  voltage  curves  OAiBi 
and  QA<iBi  (Fig.  27)  where  <fe  is  less  than  <£i.  If  it  is  not  necessary 
for  the  filament  to  give  a  greater  saturation  current  than  that  given 
by  AiBi  the  filament  with  the  smaller  value  of  </>  still  offers  a  great 
advantage  because  in  such  case  we  could  operate  the  filament  at  a 
lower  temperature.  The  power  saved  in  lowering  the  temperature 
could  then  be  used  in  increasing  the  length  of  the  filament.  This 
would  increase  the  total  space  current  and  the  characteristic 
would  take  the  form  OHB\.  It  will  be  shown  later  that  the 
steepness  of  the  characteristic  is  a  very  important  factor 
in  determining  the  efficiency  of  thermionic  amplifiers,  detectors, 
etc.  The  curve  OH  is  therefore  more  suitable  than  OA\. 

It  is  to  be  seen,  therefore,  that  it  is  very  desirable  to  use  a 
filament  with  as  low  an  electron  affinity  as  possible.  This  is 


80 


THERMIONIC  VACUUM  TUBE 


obtained  in  the  Wehnelt 1  cathode,  which  consists  of  a  platinum 
filament  coated  with  an  oxide  of  the  alkaline  earths. 

The  type  of  filament  used  in  Western  Electric  tubes  is  the 
result  of  efforts  to  reduce  the  electron  affinity.     A  comparison  of 


Anode  Volts 
FIG.  27. 


table  II  for  tungsten  filaments  with  the  following,  which  gives  the 
values  for  a  type  of  Western  Electric  filament,2  shows  the  relative 
thermionic  efficiencies  of  the  two  types. 

TABLE  IV 


p 

Is 

s=- 

P 

Watts  per  Cm2. 

Mils  per  Cm2. 

Mils  per  Watt. 

4 

11 

2.7 

5 

35 

7 

6 

80 

13 

7 

160 

23 

8 

300 

37.5 

9  . 

500 

55.5 

10 

750 

75 

1  A.  WEHNELT,  Ann.  d.  Phys.,  Vol.  14,  p.  125,  1904. 

2  From  measurements  of  C.  J.  DAVISSON. 


PHYSICS  OF  THE  THERMIONIC  VALVE  81 

The  values  for  p,  s  and  /s  given  in  this  table  also  obey  equations 
(24)  and  (25)  but  the  constants  n  and  C  for  this  filament  have  the 
values:  n  =  3.59  and  C  =  2.148Xl(r2. 

It  is  on  account  of  the  high  thermionic  efficiency  that  the 
oxide-coated  platinum  filament  can  be  operated  at  such  low  tem- 
peratures. These  filaments  should  never  be  heated  above  a  reddish 
yellow  (which  corresponds  approximately  to  p  =  8  to  9  watts  per 
cm.2),  whereas  a  tungsten  filament  can  be  heated  to  brilliancy. 

The  experimental  verificaton  of  Richardson's  equation  for 
the  Wehnelt  type  of  filament  presents  greater  difficulties  than  in 
the  case  of  pure  metallic  filaments.  For  metallic  filaments  this 
equation  was  verified  by  Richardson  in  1903,1  and  subsequently 
by  several  others.  Accurate  determinations  of  Richardson's 
constants  for  tungsten,  molybdenum  and  other  metals  were  made 
in  the  laboratories  of  the  General  Electric  Company.  The  earliest 
experiments  that  were  made  to  determine  the  electron  affinity 
for  oxide-coated  filaments  were  those  of  Wehnelt.2  Richardson's 
equation  has  been  fully  verified  for  oxide-coated  filaments  by 
investigations  carried  on  in  the  research  laboratories  of  the 
Western  Electric  Company.  Some  of  this  work  is  described 
by  H.  D.  Arnold.3 

The  coated  type  of  filament  has  now  been  used  by  the  Western 
Electric  Company  since  1913,  and  has  been  in  commercial  use  in 
the  telephone  repeater  tubes  of  the  Bell  Telephone  System  since 
1914.  This  filament  is  sufficiently  constant  in  its  behavior  to  meet 
the  very  rigid  requirements  called  for  by  its  use  on  the  long  dis- 
tance telephone  lines. 

It  consists  of  a  core  of  platinum-iridium  (6  per  cent  iridium 
with  other  impurities  found  in  commercial  platinum-iridium) 
covered  with  the  oxides  of  barium  and  strontium.  These  oxides 
are  applied  alternately  and  after  each  application  the  filament  is 
momentarily  raised  to  a  temperature  of  about  1000°  C.  The 
whole  process  consists  of  sixteen  such  applications.  After  that 
the  filament  is  baked  at  about  1200°  C.  for  two  hours.  If  the 

1  O.  W.  RICHARDSON,  Trans.  Roy.  Soc.,  Vol.  A-201,  p.  497,  1903. 

2  A.  WEHNELT,  Ann.  d.  Phys.,  Vol.  14,  p.  425,  1904.     For  a  full  discussion 
of  these  and  similar  experiments,  see  O.  W.  RICHARDSON,  "  The  Emission  of 
Electricity  from  Hot  Bodies  "  (Longmans,  London). 

3H.  D.  ARNOLD,  paper  read  at  the  Chicago  meeting  of  the  American 
Physical  Society,  October,  1919. 


82 


THERMIONIC  VACUUM  TUBE 


filament  is  not  exposed  to  moisture  or  carbon  dioxide  it  does  not 
deteriorate.  If  kept  in  vacuum  containers  they  show  no  deterio- 
ration over  a  period  of  several  years.  The  tubes  containing  these 
filaments  are  completely  interchangeable  even  in  repeater  circuits 
where  the  requirements  are  held  within  very  close  limits. 


Dower  i 


FIG.  28. 

The  investigation  on  the  thermionic  efficiency  of  the  filaments 
was  simplified  by  a  coordinate  system  devised  by  Dr.  C.  J.  Davis- 
son,  in  which  the  abscissae  represent  power  supplied  to  the  fila- 
ment, and  the  ordinates  the  thermionic  emission.  The  abscissae  of 
this  system  are  curved,  the  coordinate  lines  being  so  proportioned 


PHYSICS  OF  THE  THERMIONIC  VALVE 


83 


that  if  the  emission  of  the  filament  satisfies  Richardson's  equation, 
and  the  thermal  radiation  the  Stefan-Boltzmann  law,  the  relation 
between  the  thermionic  emission  and  the  power  supplied  to  the 
filament  when  plotted  on  this  chart  is  a  straight  line.  Such  a 
chart  is  shown  in  Fig.  28.  The  lines  represent  the  average  ther- 
mionic emission  for  a  large  number  of  different  filaments.  These 
filaments  all  have  the  same  area,  namely,  95  sq.  mm.  The  further 
the  line  lies  to  the  left,  the  greater  is  the  thermionic  efficiency. 
Each  line  shows  the  percentage  of  tubes  that  have  a  higher  ther- 
mionic emission  than  that  indicated  by  the  line.  For  most  pur- 
poses it  is  necessary  only  to  insure  that  the  thermionic  emission 
is  greater  than  a  certain  value.  The  thermionic  efficiency  is 

obtained  by  dividing    the  ordinates  by  the  abscissae    ($  =— j. 

The  broken  lines  represent  the  lines  of  constant  thermionic 
efficiency,  the  corresponding  thermionic  efficiencies  indicated  on 
this  line  being  expressed  in  milliamperes  per  watt.  The  normal 
power  dissipated  in  this  standard  coated  filament  is  from  8  to  9 
watts  per  square  centimeter.  From  this  it  is  seen  that  the 
efficiency  of  these  filaments  range  from  about  10  to  100  milli- 
amperes per  watt. 

The  constants  of  Richardson's  equation   \Is=ATl/2e  T)  can 
be  determined  directly  from  these  lines. 

The  following  table  gives  the  constants  a  and  b  of  Richardson's 
equation  for  a  number  of  different  substances.  The  values  for 
the  Western  Electric  oxide-coated  filament  were  obtained  by  C.  J. 
Davisson  from  measurements  covering  about  4000  filaments.1 
The  values  for  the  other  substances  given  in  the  table  are  taken 
from  a  paper  by  Langmuir.2 

TABLE  V 


Substance. 

Amps/Cm2. 

b 
Kelvin  Degrees. 

Oxide  coat  (W.  E.  Standard). 
Tungsten  

(8-24)Xl04 
2.36X107 

(1.94-2.38)X104 
5.25X104 

Thorium      

2.0X108 

3.9  X104 

Tantalum                       

1.12X107 

5.0X104 

Molybdenum 

2.1X107 

5.0X104 

1  H.  D.  ARNOLD,  loc.  cit. 

2 1.  LANGMUIR,  Trans.  Am.  Electrochem.  Soc.,  Vol.  29,  p.  138,  1916. 


84  THERMIONIC  VACUUM  TUBE 

The  thermionic  efficiency  is  determined  mainly  by  6;  the  smaller 
6  the  greater  the  efficiency. 

To  obtain  the  electron  affinity  0  the  equation  <£  =  8.6X10~5X& 
(Chapter  III,  equation  (9))  can  be  used. 

33.  Life  of  a  Vacuum  Tube.  The  life  of  a  tube  is  determined 
mainly  by  two  factors: 

(a)  There  is  always  a  small  amount  of  ionization  by  collision 
even  in  highly  evacuated  tubes.     The  positive  ions  so  formed 
bombard  the  filament  and  this  causes  excessive  local  heating. 
In  the  three-electrode  type  of  tube  the  grid  acts  as  a  partial 
screen  to  positive  ion  bombardment.     The  electric  field  in  the 
region  between  grid  and  plate  is  usually  much  greater  than  between 
grid  and  filament.     Most  of  the  ionization,  therefore,  takes  place 
between  grid  and  plate  and  a  large  percentage  of  the  resulting 
positive  ions  go  to  the  grid  instead  of  to  the  filament,  since  the  grid 
is  always  negative  with  respect  to  the  plate. 

(b)  The  rate  at  which  the  filament  volatilizes  increases  with 
its  temperature.     In  the  case  of  the  metallic  filaments,  the  vola- 
tilization causes  the  filament  gradually  to  get  thinner  and  so  in- 
creases its  resistance.     If  the  filament  is  operated  at  constant 
voltage  this  will  cause  a  reduction  in  the  heating  current,  and  the 
consequent  lowering  of  the  temperature  lowers  the  thermionic 
emission  as  well  as  the  thermionic  efficiency.     If  the  filament  is 
operated  at  constant  current  the  voltage  increases,  resulting  in  an 
increase  of  the  temperature  of  the  filament.     This  shortens  the 
life  of  the  filament.     Whether  the  filament  be  operated  at  constant 
voltage  or  constant  current,  both  effects  are  undesirable  and  must 
be  taken  into  consideration  in  estimating  the  life  of  the  filament. 

The  life  of  a  metallic  filament  depends  also  on  its  diameter.1 
A  5-mil  tungsten  filament  operated  at  a  temperature  of  2400°  K. 
has  a  life  of  about  4000  hours,  while  the  10-mil  filament  operated 
at  2500°  K.  has  a  life  of  nearly  3000  hours.  The  thicker  the 
filament  the  longer  the  life  for  the  same  operating  temperature. 
Or,  the  same  length  of  life  can  be  obtained  by  operating  the 
thicker  filament  at  a  higher  temperature  and  so  obtain  a  greater 
thermionic  emission,  as  well  as  a  higher  thermionic  efficiency, 
since  the  thermionic  efficiency  increases  with  the  temperature. 

The  following  table  taken  from  Dushman's  paper  gives  an 
idea  of  the  effect  of  the  diameter  of  the  filament  on  its  life : 
1  S.  DUSHMAN,  General  Electric  Review,  Vol.  18,  p.  156,  1915. 


PHYSICS  OF  THE  THERMIONIC  VALVE  85 

TABLE  VI 


Filament 

Safe  Temperature 

Is  per 

Watts  per 

Diameter,  Mils 

(Life>2000Hrs.). 

Cm.  length. 

Cm.  Length. 

5 

2475 

30 

3.1 

7 

2500 

50 

4.6 

10 

2550 

100 

7.2 

15 

2575 

200 

11.3 

By  "  safe  temperature  "  here  is  meant  a  temperature  which 
is  low  enough  to  insure  a  life  of  at  least  2000  hours.  The  quanti- 
ties given  in  the  third  column  give  the  thermionic  emission  per 
centimeter  length  of  filament  at  the  corresponding  temperature, 
and  the  fourth  column  gives  the  power  that  must  be  expended  in 
maintaining  a  centimeter  length  of  the  filament  at  that  tempera- 
ture. The  thermionic  efficiency  can  be  obtained  by  dividing  the 
values  in  the  third  column  by  those  in  the  fourth.  It  is  seen  that 
the  thermionic  efficiency  of  the  15-mil  filament  is  almost  twice 
that  of  the  5-mil  filament  when  both  are  operated  at  such  tempera- 
tures as  to  give  approximately  the  same  life. 

The  coated  type  of  filament  retains  a  constant  resistance 
throughout  its  life,  because  the  heating  current  in  this  filament 
is  carried  mainly  by  the  core,  while  what  evaporates  is  mostly  the 
coating.  The  nearing  of  the  end  of  this  filament  is  indicated  by 
an  increase  in  the  temperature  over  sections  of  its  length.  These 
are  commonly  referred  to  as  "  bright  spots."  This  warning  is  a 
desirable  and  important  feature,  especially  where  the  tube  is  used 
as  a  telephone  repeater,  because  it  makes  possible  a  timely  replace- 
ment of  the  tube  without  interrupting  the  service. 

Tubes  containing  the  standard  Western  Electric  filament 
have  a  life  of  several  thousand  hours,  which  depends,  of  course, 
upon  the  temperature  at  which  the  filament  is  operated.  Such 
tubes  have  been  operated  in  the  laboratory  for  20,000  hours  con- 
tinuously, during  which  period  the  thermionic  current  remained 
practically  constant. 


CHAPTER  V 
INFLUENCE  OF  GAS  ON  THE  DISCHARGE 

THE  discussion  given  in  the  previous  chapter  and  the  current 
voltage  relations  that  were  obtained  were  based  on  the  assumption 
that  the  residual  gas  in  the  device  has  a  negligibly  small  influence 
on  the  discharge.  It  now  remains  to  show  under  what  conditions 
this  assumption  is  justified  and  how  these  conditions  can  be  realized 
in  practice.  It  is  important  to  know  what  are  the  sources  cf  gas 
in  thermionic  tubes;  how  the  gas  influences  the  discharge  and  how 
the  c'eleterious  effects  of  gas  can  be  eliminated. 

There  are  two  principal  ways  in  which  the  presence  of  gas  in  a 
thermionic  tube  can  affect  the  discharge.  Firstly,  gas  in  contact 
with  the  surface  of  a  cathode  can  change  the  thermionic  emission 
from  the  cathode  and  so  change  the  saturation  current,  i.e.,  the 
total  current  obtainable  from  it  at  a  definite  temperature.  This 
effect  may  be  referred  to  as  the  surface  effect.  Secondly,  the  pres- 
ence of  gas  in  the  space  between  cathode  and  anode  will,  if  the 
velocity  of  the  electrons  coming  from  the  cathode  exceeds  a  certain 
small  value,  depending  on  the  nature  of  the  gas.  give  rise  to  the 
phenomencn  cf  ionization  by  collision.  This  can  be  referred  to  as 
the  volume  effect. 

34.  Volume  Effect  of  Gas.  Ionization  by  Collision.  In  order 
to  explain  the  effect  of  ionization  by  collision  on  the  discharge,  we 
shall  assume  that  we  have  a  characteristic  corresponding  to  that 
obtained  in  a  perfect  vacuum  and  then  see  how  this  characteristic 
is  changed  when  gas  to  a  sufficiently  high  pressure  is  introduced 
into  the  tube.  We  shall  also  acsume  that  the  gas  which  is  intro- 
duced is  entirely  neutral  as  regards  the  surface  effect;  that  is, 
it  is  of  such  a  nature  that  its  coming  in  contact  with  the  surface 
cf  the  cathode  does  not  change  the  electron  emission  from  the 
cathode.  In  passing  from  cathode  to  anode,  some  of  the  electrons 
collide  with  the  molecules  of  the  gas  and  if  they  strike  the  mole- 
cules with  a  velocity  exceeding  a  definite  minimum  amount  ioniza- 

80 


INFLUENCE  OF  GAS  ON  THE  DISCHARGE  87 

tion  by  collision  sets  in.  The  voltage  through  which  an  electron 
must  drop  to  acquire  this  minimum  velocity  is  called  the  ioniza- 
tion  voltage,  values  of  which  are  given  on  page  22. 

If  the  voltage  between  cathode  and  anode  is  slightly  greater 
than  the  ionization  voltage,  then,  if  an  electron  collides  with  a  gas 
molecule  just  before  reaching  the  anode,  ionization  will  result,  but  a 
collision  in  the  spa.ce  nearer  to  the  cathode  will  not  result  in  ioniza- 
tion. In  the  latter  case  the  electron  may  be  reflected  without 
any  loss  of  energy  from  the  molecule  with  which  it  collides,  or  it 
may  lose  part  or  all  of  its  energy,  this  energy  being  transferred 
to  the  molecule,  or  it  may  combine  with  the  molecule,  thus  forming 
a  heavy  negative  carrier.  It  can  readily  be  seen  that  if  the  voltage 
between  .cathode  and  anode  be  increased,  collision  of  the  electrons 
with  molecules  nearer  to  the  cathode  may  result  in  ionization,  and 
if  the  voltage  just  exceeds  twice  the  ionization  voltage,  an  electron 
which  collides  after  having  dropped  through  the  ionizaticn 
voltage  in  moving  from  the  cathode,  thus  producing  ionization, 
stands  a  chance  of  ionizing  another  molecule  with  which  it  may 
happen  to  collide  just  before  reaching  the  anode.  For  low  volt- 
ages, therefore,  it  is  to  be  expected  that  the  amount  of  ionization 
would  increase  with  the  applied  voltage. 

In  practice  we  do  not  deal  with  a  single  electron  moving  from 
cathode  to  anode  but  with  a  stream  of  electrons,  and  under  such 
conditions  it  is  generally  found  that  ionization  sets  in  at  applied 
voltages  less  than  the  ionization  voltage.  It  is,  for  example, 
possible  to  maintain  an  arc  in  a  gas  or  vapor  by  the  application 
of  a  voltage  which  is  not  as  great  as  the  ionization  voltage  of  the 
gas  or  vapor.  This  is  because  it  takes  a  smaller  amount  of  energy 
to  disturb  the  equilibrium  of  an  atom  than  it  does  to  completely 
detach  an  electron  from  an  atom.  Once  the  equilibrium  of  an 
atom  has  been  disturbed  the  potential  energy  of  the  atomic 
system  is  increased  by  an  amount  equal  to  the  energy  given  up  to 
the  atom  by  the  colliding  electron.  Such  an  atom  is  more  easily 
ionized  than  the  normal  atom  and  therefore  the  potential  differ- 
ence through  which  any  electron  must  drop  in  order  to  ionize  this 
atom  is  less  than  the  ionization  voltage  of  the  normal  atom.1 

The  amount  of  ionization  depends  also  on  the  pressure  of  the 
gas.     The  pressure  of  the  gas  may  be  so  low  that  the  electron  does 
not  strike  a  molecule  at  all  in  its  flight  from  cathode  to  anode.     On 
i  H.  J.  VAN  DER  BIJL,  Phys.  Rev.,  Vol.  10,  p.  546,  1917. 


88 


THERMIONIC  VACUUM  TUBE 


the  other  hand,  the  pressures  may  be  so  high  that  the  electron 
collides  before  it  has  acquired  sufficient  energy  to  ionize.  The 
amount  of  ionization  produced  in  this  case  depends  on  whether  or 
not  the  gas  is  such  that  the  collisions  are  elastic.  If  they  are 
elastic  the  electrons  will  rebound  from  the  molecules  without  losing 
their  energy  and  may  then  strike  the  next  molecules  with  a  greater 
amount  of  energy  than  the  first.  If  the  collisions  are  inelastic 
the  electrons  lose  some  or  all  of  their  energy  on  colliding,  but  the 
energy  which  is  transferred  to  the  molecules  is  again  radiated  from 
them  in  the  form  of  light,  which  causes  photo-electric  effects 
in  the  tube,  resulting  in  a  further  dislodgment  of  electrons. 

35.  Mean  Free  Path  of  Electrons  in  Gases.  The  chance  that 
an  electron  has  of  colliding  with  a  gas  molecule  in  its  passage  from 
cathode  to  anode  depends  on  the  mean  free  path  of  the  electrons  in 
the  gas  and  upon  the  distance  between  cathode  and  anode.  The 
mean  free  path  is  the  average  distance  through  which  an  electron 
can  move  freely  without  colliding  with  gas  molecules.  The 
following  table  gives  an  idea  of  the  nature  of  this  important  quan- 
tity. The  first  column  gives  the  number  N  of  electrons,  out  of  a 
total  of  100  starting  from  the  cathode,  that  can  move  freely  through 
the  distance  d  given  in  the  second  column  of  the  table.  The 
numbers  in  the  second  column  are  expressed  in  fractions  of  the 
mean  free  path  L. 


2V 

d 

~L 

99 

0.01 

98 

0.02 

90 

0.1 

82 

0.2 

78 

0.25 

72 

0.333 

61 

0.5 

37 

1. 

14 

2 

5 

3 

2 

4 

1 

4.6 

This  table  snows  that  if  the  distance  between  cathode  and 
anode  is  equal  to  the  mean  free  path,  only  37  per  cent  of   the 


INFLUENCE  OF  GAS  ON  THE  DISCHARGE  89 

electrons  starting  from  the  cathode  will  strike  the  anode  without 
having  encountered  molecules  on  their  way,  and  as  the  ratio  of  the 
distance  between  cathode  and  anode  to  the  mean  free  path  is 
increased  the  number  of  collisions  increases.  On  the  other  hand, 
if  the  pressure  is  so  low  that  the  mean  free  path  is  100  times  the 
distance  between  cathode  and  anode,  only  1  per  cent  of  the 
electrons  will  collide  before  reaching  the  anode. 

The  mean  free  path  increases  as  the  gas  pressure  is  decreased; 
in  fact,  it  is  inversely  proportional  to  the  pressure  of  the  gas.  It 
also  depends  upon  the  size  of  the  molecules.  Thus  it  is  greater  for 
hydrogen  than  for  oxygen.  Consequently,  since  an  electron  is 
much  smaller  than  a  gas  molecule,  the  mean  free  path  of  electrons 
in  gases  is  greater  than  the  mean  free  path  of  the  gas  molecules 
themselves.  In  order  to  obtain  the  mean  free  path  of  electrons 
in  the  gas  from  the  mean  free  path  of  the  gas  molecules  in  the  gas 
itself,  we  must  multiply  by  the  factor  4V2.  If  L  is  the  mean 
free  path  of  the  gas  molecules  at  atmospheric  pressure  (760  mm. 
of  Hg),  then  the  mean  free  path  of  elections  in  that  gas,  at  a  pres- 
sure p  is: 

~,      «   •   •    ,r.;  ......'    ..-..     . 

where  p  is  given  in  millimeters  of  Hg. 

The  mean  free  path  for  most  of  the  common  gases  is  given  in 
tables  of  physical  constants.1  ' 

The  mean  free  path  of  a  gas  or  vapor  can  be  obtained  if  the 
coefficient  of  viscosity  is  known.  The  coefficient  of  viscosity  is 

given  by  the  equation 

Q  -t   —  T  9  f^y\ 

where 

rj  =  coefficient  of  viscosity; 
p  =  density  of  gas; 
c  =  mean  molecular  velocity; 
L  =  mean  free  path; 

the  quantities  being  reduced  to  atmospheric  pressure.     Now,    the 
pressure  P  is  given  by 

3/3P (3) 

/.  C  =  A/— 
\  p    J 

lSee,  for  example,  "Physical  and  Chemical  Constants,"  by  G.  W.  C. 
Kaye  and  T.  H.  Laby, 


90  THERMIONIC  VACUUM  TUBE 

The  mean  free  path  for  any  gas  for  which  the  coefficient  of 
viscosity  is  known  can  be  obtained  readily  from  the  known  mean 
free  path  and  coefficient  of  viscosity  of  some  other  gas.  For 
example,  from  equations  (2)  and  (3)  we  obtain 


(4) 


and  therefore  if  Z/2  is  the  known  mean  free  path  of  one  gas  at  atmos- 
pheric pressure,  the  mean  free  path  LI  for  the  other  gas  at  the 
same  pressure  is  given  by 


where  M  i  and  MI  are  the  molecular  weights  of  the  two  gases  con- 
sidered. The  mean  free  path  of  the  electrons  in  the  gas  at  some 
other  pressure  can  -then  be  obtained  from  equation  (1). 

36.  lonization  at  Low  Pressures.  The  application  of  the 
theory  of  ionization  by  collision  when  the  pressure  is  of  such  order 
of  magnitude  that  the  mean  free  path  is  large  compared  to  the  dis- 
tance between  the  electrodes  is  simpler  than  when  the  mean  free 
path  is  of  the  same  order  as,  or  less  than,  the  electrode  distance. 
The  relation  between  ionization  current  and  the  pressure  is  also 
simpler.  Let  us  consider  the  case  in  which  the  pressure  of  the  gas 
in  the  tube  is  so  low  that  the  mean  free  path  is  large  compared 
with  the  distance  between  cathode  and  anode.  If  p  is  the  pres- 
sure in  millimeters  of  Hg  and  N  the  number  of  gas  molecules  per 
cubic  centimeter  at  atmospheric  pressure,  the  number  of  mole- 

vN 

cules  per  cubic  centimeter  at  the  pressure  p  is  |—.  . 

i  t)U 

Let  us  suppose  that  cathode  and  anode  are  both  in  the  form  of 
infinitely  large  parallel  plates,  and  let  the  number  of  electrons 
moving  away  from  1  square  centimeter  of  cathode  surface  per 
second  be  n\.  In  moving  from  cathode  to  anode  some  of  these 
electrons  will  collide  with  the  gas  molecules.  If  the  voltage 
between  cathode  and  anode  be  so  high  that  every  collision  results 
in  ionization,  the  member  of  positive  ions  formed  will  be  equal 
to  the  number  of  collisions,  and  since  the  mean  free  path  is  large 
compared  with  the  electrode  distance,  the  chance  of  an  electron 
colliding  more  than  once  on  its  way  to  the  anode  will  be  extremely 
small.  We  can  therefore  imagine  the  molecules  in  the  space 


INFLUENCE  OF  GAS  ON  THE  DISCHARGE  91 

projected  on  the  plane  of  the  anode  and  compute  the  ratio 
of  the  area  covered  by  the  molecules  to  area  of  the  anode.  This 
will  then  be  proportional  to  the  ratio  of  the  positive  ions  formed 
by  collision  to  the  number  of  electrons  moving  from  cathode  to 

anode.     Now,  the  cross-sectional  area  formed  by  the  —^  mole- 


o      TI  T 

cules  is  -zr  where  r  is  the  molecular  radius.     Hence,  if  712  be 


the  number  of  positive  ions  resulting  from  collisions  in  a  column 
1  square  centimeter  in  cross-section,  and  n\  the  number  of  elec- 
trons moving  to  the  anode  per  second  from  1  square  centimeter 
of  cathode  surface,  we  have 

H2  _  1  Trr2pN 

n[~k  ~760~ 
or 

760/c   n2  ,R. 

P  —  -  OA>-  -  ,        .....        .        .        (6) 

irr2N  ni 

where  k  is  a  constant  which  becomes  unity  if  all  the  molecules  in 
the  path  of  the  electron  stream  are  ionized. 

The  pressure  p  of  the  gas  is  therefore  directly  proportional 
to  the  ratio  of  positive  ions  to  electrons.  This  linear  relation  has 
been  observed  experimentally  by  O.  E.  Buckley  1  and  on  the  basis 
of  this  simple  relationship  redesigned  a  thermionic  gauge  for  the 
measurement  of  pressures  below  about  10  ^3  mm.  of  Hg.  This 
gauge  is  described  in  Chapter  X,  page  375. 

37.  Effects  of  lonization  by  Collision.  When  ionization  takes 
place,  the  characteristic  can  be  influenced  in  the  following  ways: 

(a)  The  splitting  of  the  gas  molecules  by  bombardment  of 
the  electrons,  in  their  passage  from  cathode  to  anode,  results  in 
the  production  of  more  dislodged  charges  so  that  the  current  is 
increased.  This  increase  in  current  is  small  under  the  conditions 
prevailing  in  most  thermionic  tubes.  Thus,  if  the  pressure  in  the 
tube  is  0.1  micron,2  the  increase  in  current  due  to  this  cause  alone 
is  less  than  1  per  cent. 

(6)  The  positive  ions  resulting  from  the  collisions  move 
toward  the  cathode  and  since  the  total  space  charge  is  the  differ- 
ence between  that  due  to  the  electrons  and  that  due  to  the  positive 
ions,  the  presence  of  the  positive  ions  naturally  reduces  the  total 

1  O.  E.  Buckley,  Proc.  National  Academy  of  Science,  Vol.  2,  p.  683,  1916. 

2  1  micron  =  10~3  mm.  of  Hg. 


92  THERMIONIC  VACUUM  TUBE 

space  charge,  and  this  causes  an  increase  in  the  current.  The 
extent  to  which  the  space  charge  of  the  positive  ions  can  reduce 
the  negative  space  charge  of  the  electrons  depends  on  the  number 
of  positive  ions  compared  with  the  number  of  electrons  in  the  space 
at  any  particular  time.  It  depends,  therefore,  on  the  speed  with 
which  the  positive  ions  move  toward  the  cathode;  the  lower  the 
speed  the  greater  will  be  the  density  of  the  positive  ions.  Thus 
when  the  tube  contains  oxygen  the  reduction  in  the  negative  space 
charge  is  greater  than  would  be  the  case  if  the  tube  contained 
hydrogen  at  such  a  pressure  that  the  number  of  positive  ions  formed 
is  the  same,  because  the  oxygen  ions  are  heavier  and  move  more 
slowly  than  the  hydrogen  ions. 

(c)  The  positive  ions  can,  under  certain  conditions,  combine 
with  the  electrons  at  the  surface  of  the  cathode  and  so  form  a  layer 
of  gas  on  it.     This  results  in  a  surface  effect  which  will  be  explained 
in  Section  39. 

(d)  There  is  still  another  way  in  which  ionization  by  collision 
can  affect  the  operation  of  the  tube.     If  the  voltage  between 
cathode  and  anode  is  sufficiently  high,  the  bombardment  of  the 
cathode  by  the  positive  ions  causes  an  abnormal  heating  of  the 
cathode.     This  increases  the  saturation  current   because  of   the 
increase  in  the  temperature  of  the  cathode.     There  may  also  be  a 
direct  emission  of  electrons  from  the  cathode  under  the  bombard- 
ment of  its  surface  by  the  positive  ions.     The  undue  heating  caused 
by  the  bombardment  wears  away  the  cathode  and  has  a  very 
deleterious  effect  on  the  life  of  the  tube. 

(e)  When  the  velocity  of  the  electrons  is  less  than  the  value 
necessary  to  cause  ionization  by  collision,  the  electrons  attract 
the   neutral  gas  molecules  and  so  form  heavy  negative  carriers. 
The  ease  with  which  this  formation  of  negative  carriers  takes  place 
depends  on  the  nature  of  the  gas.     Such  gases  as  argon  and  mer- 
cury vapor  do  not  readily  form  negative  carriers,  while  hydrogen 
and  oxygen  combine  with  electrons  more  easily.     The  effect  of 
this  negative  carrier  formation  is  to  counteract  the  reduction  in 
the  negative  space  charge  occasioned  by  the  heavy  positive  ions 
formed  by  collision  ionization.     The  positive  ions  are  atoms  of  the 
gas  from  which  one  or  more  electrons  have  been  removed.     The 
ions  therefore  have  very  nearly  the  same  weight  as  the  gas  atoms. 
The  negative  carriers,  on  the  other  hand,  may  consist  of  an  atom 
or  molecule  to  which  has  been  attached  an  electron.     It  is  also 


INFLUENCE  OF  GAS  ON  THE  DISCHARGE  93 

possible  that  the  attraction  between  an  electron  and  the  neutral 
gas  molecules  can  result  in  the  formation  of  clusters  consisting 
of  more  than  one  molecule  held  together  by  the  electron.  These 
negative  carriers,  therefore,  move  as  slowly  as,  and  sometimes  more 
slowly  than,  the  positive  ions  and  consequently  have  a  relatively 
great  effect  in  counteracting  the  tendency  of  the  positive  ions  to 
reduce  the  negative  space  charge  of  the  electrons. 

38.  Influence  of  lonization  on  the  Infra-Saturation  Part  of  the 
Characteristic.  To  confine  our  attention  to  this  part  of  the  char- 
acteristic, let  us  assume  that  the  tube  contains  gas  which  has  no 
effect  on  the  electron  affinity  of  the  cathode;  in  other  words,  it 
has  no  effect  in  either  reducing  or  increasing  the  saturation  current 
obtainable  from  the  filament  at  any  definite  temperature.  This 
condition  can  be  realized,  for  example,  with  a  tube  containing  a 
tungsten  cathode  and  mercury  vapor  because  mercury  vapor  is 
neutral  as  regards  the  electron  emission  from  tungsten.1  Let 
us  suppose  that  we  measure  the  current  voltage  relation  of  a  tube 
containing  a  tungsten  cathode  and  a  tungsten  anode,  and  to  which 
is  attached  an  appendix  containing  mercury  which  can  be  main- 
tained at  any  desired  temperature  thus  maintaining  the  pressure 
of  the  mercury  vapor  at  any  desired  value.  It  is,  of  course,  to  be 
understood  that  all  other  gases  and  vapors  have  been  driven  out 
of  the  electrodes  and  the  walls  of  the  vessel.  It  will  be  appre- 
ciated that  by  doing  this  we  do  not  simulate  the  conditions  ob- 
taining in  practical  thermionic  tubes,  because  the  gases  remaining 
in  the  practical  tubes  seldom,  if  ever,  contain  mercury  vapor,  but 
do  comprise  those  gases  that  we  are  excluding  from  the  experi- 
ment at  present  under  consideration.  For  such  an  experiment 
it  is  desirable  to  use  a  vapor  which  can  be  maintained  at  any 
pressure  by  keeping  the  parent  substance  at  the  corresponding 
temperature. 

Let  us  first  suppose  the  appendix  containing  the  mercury  is 
immersed  in  liquid  air.  Under  such  conditions  the  characteristic 
will  be  that  obtained  in  a  high  vacuum.  Then,  to  study  the  effect 
of  the  mercury  vapor  in  increasing  the  current  on  the  infra-satura- 
tion part  of  the  characteristic  we  can,  instead  of  using  liquid  air, 
maintain  the  mercury  at  other  temperatures  by  dipping  the  mer- 
cury tube  into  freezing  mixtures  or  water  baths.  A  set  of  such 
characteristics  is  shown  in  Fig.  29.  The  characteristic  marked 
1  IRVING  LANGMUIR,  Physik.  Zeitsch.,  Vol.  15,  p.  519,  1914. 


94 


THERMIONIC  VACUUM  TUBE 


No.  1  was  obtained  with  the  mercury  tube  immersed  in  liquid  air. 
The  other  characteristics  were  obtained  with  the  mercuiy  at 
temperatures  ranging  from  —7.5°  C.  to  10°  C.  The  temperatures 
below  0°  C.  were  obtained  by  keeping  the  tube  containing  the 


40       (     50  60 

Anode  VolH 

FIG.  29. 

mercury  immersed  in  ice  and  salt  freezing  mixtures.  The  following 
table  shows  the  temperatures  of  the  mercury  and  the  pressures  of 
the  mercury  vapor  corresponding  to  the  set  of  characteristics 
shown  in  Fig.  29. 

TABLE  VII 


Curve  No. 

Temperature  of  Hg, 
°C. 

Pressure  of  Hg  Vapor, 
Micron. 

1 

-185 

0.000 

2 

-7.5 

0.085 

3 

0.0 

0.18 

4 

5.0 

0.3 

5 

7.5 

0.4 

6 

10.0 

0.5 

The  pressures  of  the  mercury  vapor  at  these  temperatures  are 
obtained  from  a  table  given  by  Knudsen.1     Referring  to  this  table 
1  M.  KNUDSEN,  Ann.  d.  Phys.,  Vol.  29,  p.  179,  1909. 


INFLUENCE  OF  GAS  ON  THE  DISCHARGE  95 

and  to  Fig.  29,  it  will  be  seen  that  as  the  pressure  of  the  mercury 
vapor  is  increased  the  current  voltage  curve  becomes  steeper  and 
steeper.  When  the  pressure  rises  to  a  value  of  0.5  of  a  micron, 
the  current  shows  a  rather  sudden  increase  at  about  33  volts 
(curve  6).  Below  this  pressure  there  is  no  sudden  increase  in 
current  on  the  infra-saturation  part.  For  a  still  higher  pressure  of 
the  mercury  vapor,  about  2  microns  or  more,  the  current 
increases  sharply  at  about  15  to  20  volts,  as  indicated  by  the 
dotted  curve  in  Fig.  29. 

At  these  pressures  of  the  mercury  vapor,  the  formation  of 
the  positive  ions  by  collision  of  the  electrons  with  the  mercury 
molecules  neutralizes  the  space  charge  of  the  negative  electrons 
so  that  the  current  that  can  flow  through  the  tube  undergoes  a 
considerable  increase.  For  this  neutralization  of  the  negative 
space  charge  it  is  necessary,  firstly,  that  the  potential  difference 
between  cathode  and  anode  be  high  enough  to  cause  a  considerable 
amount  of  positive  ionization.  In  the  case  of  mercury  vapor, 
this  voltage  is  of  the  order  of  5  volts  or  less.  Secondly,  it  is  neces- 
sary that  the  number  of  positive  ions  formed  by  collision  be  great 
compared  with  the  number  of  negative  carriers  formed  by  com- 
bination of  the  electrons  with  the  neutral  molecules.  The  for- 
mation of  heavy  negative  carriers  accounts  at  least  partly  for  the 
fact  that  the  negative  space  charge  does  not  become  neutralized 
until  the  voltage  becomes  considerably  greater  than  the  voltage 
through  which  the  electrons  must  drop  to  produce  ionization  by 
collision.  Thus,  the  current  does  not  increase  suddenly  until  the 
filament-plate  voltage  reaches  about  15  volts  or  more,  although 
actually  a  considerable  amount  of  ionization  by  collision  takes 
place  at  much  lower  voltages.  The  reason  why  the  current  does 
not  increase  when  ionization  takes  place  at  these  low  voltages  is 
because  a  relatively  large  number  of  negative  carriers  are  formed 
by  the  combination  of  electrons  with  the  neutral  gas  molecules, 
and  these  negative  carriers  tend  to  neutralize  the  space  charge  of 
the  positive  ions.  Another  factor  wriich  tends  to  counteract  the 
reduction  of  the  space  charge  of  the  electrons  is  the  recombina- 
tion of  the  positive  and  negative  charges. 

The  extent  to  which  the  current  is  increased  by  ionization 
by  collision  depends  of  course  not  only  on  the  pressure  of  the  gas 
vapor,  but  also  on  the  distance  between  cathode  and  anode.  The 
important  quantity  that  determines  the  amount  of  collision  ioniza- 


96  THERMIONIC  VACUUM  TUBE 

tion  is  the  ratio  of  the  mean  free  path  of  the  electrons  in  the  gas 
or  vapor  to  the  electrode  distance.  Since  the  pressures  corre- 
sponding to  the  curves  shown  in  Fig.  29  are  known,  we  can  from 
these  curves  find  the  ratio  of  the  mean  free  path  to  the  electrode 
distance  for  the  maximum  pressure  at  which  there  is  no  sudden 
change  in  the  characteristic  due  to  the  presence  of  the  gas.  These 
curves  show  that  this  maximum  pressure  of  mercury  vapor  is  about 
0.4ju  for  the  tube  with  which  they  were  obtained. 

Now,  the  effects  as  shown  by  these  curves  are  much  greater 
for  mercury  vapor  than  for  the  gases  that  commonly  remain  as 
residual  gases  in  thermionic  tubes.  In  order  to  get  an  indication 
of  the  extent  to  which  the  other  gases  increase  the  current  on  the 
infra-saturation  part  of  the  characteristic,  we  have  to  distinguish 
between  mercury  vapor  and  such  gases  in  three  respects:  Firstly, 
mercury  molecules  are  very  heavy  compared  to  the  molecules  of 
the  ordinary  gases,  such  as  hydrogen,  oxygen  and  nitrogen,  and, 
therefore,  have  a  greater  effect  in  reducing  the  negative  space 
charge.  Secondly,  the  mean  free  path  of  electrons  in  mercury 
vapor  seems  to  be  considerably  less  than  the  mean  free  path  of  elec- 
trons in  the  common  gases.  Thirdly,  such  gases  as  oxygen  and 
hydrogen  show  a  greater  tendency  to  form  negative  carriers 
by  combining  with  electrons  than  mercury  vapor.  It  is  also 
possible  that  the  rate  of  recombination  of  the  positive  ions  with 
electrons  is  different  for  different  gases. 

In  attempting  to  obtain  an  indication  of  the  minimum  value  of 
the  ratio  of  mean  free  path  to  the  electrode  distance  necessary  to 
give  a  characteristic  which  on  the  infra-saturation  part  is  not 
influenced  to  a  disturbing  extent  by  the  presence  of  gases  other 
than  mercury  vapor,  the  differences  mentioned  above  must  be  con- 
sidered. Little  is  known  with  regard  to  the  difference  in  the 
coefficient  of  recombination  or  the  rate  of  formation  of  negative 
carriers  in  different  gases  and  vapors.  We  can,  however,  obtain 
some  indication  of  the  maximum  allowable  pressure  of  such  gases 
as  hydrogen  and  oxygen  by  considering  only  the  velocity  of  the 
ions  and  the  mean  free  path  of  the  electrons  in  the  gas. 

Suppose  that  the  minimum  value  of  the  ratio  of  mean  free  path 
to  electrode  distance  for  mercury  has  been  determined  from  a  set 
of  curves  like  that  shown  in  Fig.  29.  The  extent  to  which  positive 
ions  reduce  the  negative  space  charge  depends  upon  the  velocity 
of  the  ions  in  the  electric  field,  due  to  the  potential  difference 


INFLUENCE  OF  GAS  ON  THE  DISCHARGE  97 

between  anode  and  cathode  and  is  inversely  proportional  to  it. 
Thus,  if  l\  is  the  mean  free  path  of  the  electrons  in  mercury  vapor 
at  the  maximum  permissible  pressure,  and  d  the  distance  between 

the  electrodes,  then  the  ratio  of  -  for  any  other  gas  is 


div 


where  MI  and  M  are  the  molecular  weights  of  mercury  and  the  gas 
considered,  and  vi  and  v  are  the  corresponding  velocities  of  their 
ions  under  the  same  electro-static  field.  Take,  for  example,  the 
case  of  hydrogen  and  mercury  vapor,  since  the  molecular  weight 
of  hydrogen  is  2  and  that  of  mercury  200,  the  permissible  ratio  of 
mean  free  path  to  electrode  distance  is  one-tenth  as  great  for 
hydrogen  as  it  is  for  mercury.  This  does  not,  however,  mean  that 
when  the  tube  contains  hydrogen  the  pressure  can  be  ten  times  as 
high  as  when  it  contains  mercury  vapor;  it  can,  in  fact,  be  still 
higher  because  the  mean  free  path  of  electrons  in  hydrogen  is 
greater  than  that  of  electrons  in  mercury  vapor  at  the  same  pres- 
sure. The  relation  between  the  pressure  and  mean  free  path  is 
shown  by  equation  (1).  If  L  and  LI  be  the  mean  free  paths  of  the 
gas  and  of  mercury  vapor  at  atmospheric  pressure,  and  /  and  h 
the  corresponding  mean  free  paths  at  the  pressures  p  and  pi, 
then  we  have 


Pi       i 

Substituting  the  value  of  -r  from  equation  (7),  we  obtain: 

k 


From  this  equation  the  maximum  allowable  pressure  p  for  any 
gas  can  be  obtained  from  the  maximum  allowable  pressure  for 
mercury  vapor.  In  the  case  of  hydrogen,  for  example,  we  have 
L=18.5X10~6  cm.  at  atmospheric  pressure.  LI,  the  mean  free 
path  of  mercury  vapor  at  atmospheric  pressure,  can  be  taken 
to  be  about  6X10~6  cm.1  If  these  values  be  inserted  in  equa- 

1  If  the  mean  free  path  of  mercury  vapor  is  computed  from  equation  (5) 
by  putting  its  coefficient  of  viscosity  equal  to  162X10"6  (figure  given  by 


98  THERMIONIC  VACUUM  TUBE 

tion  (8)  we  find  that  if  the  tube  contains  hydrogen  the  maximum 
allowable  pressure  is  about  thirty  times  as  high  as  when  the 
tube  contains  mercury.  When  the  tube  contains  oxygen  or 
nitrogen  the  pressure  can  be  about  four  times  as  high  as  in  the  case 
of  mercury  vapor. 

For  the  practical  operation  of  thermionic  devices  it  is  necessary 
that  the  current  over  the  operating  range  should  not  show  erratic 
changes.  It  will  be  apparent-  from  the  previous  discussions  that 
the  pressure  necessary  to  secure  a  discharge  that  is  not  appreciably 
influenced  by  gas  is  of  such  an  order  of  magnitude  that  it  can 
readily  be  obtained.  But  it  is  important  also  to  maintain  the 
pressure  constant  enough  to  prevent  any  appreciable  changes  in 
the  effects  of  ionizatiori  on  the  characteristic.  To  secure  this  the 
electrodes  and  walls  of  the  vessel  must  be  freed  of  gas  to  such  an 
extent  that  the  heating  of  these  parts  during  the  operation  of 
the  tubes  does  not  cause  the  evolution  of  enough  gas  to  bring 
about  such  pressure  changes.  The  part  of  the  characteristic  on 
which  the  great  majority  of  thermionic  devices  operate  is  the 
infra-saturation  part  that  we  have  discussed  in  the  previous  pages. 
The  effect  of  gas  on  the  saturation  part  of  the  characteristic  will 
be  discussed  in  the  following  section. 

39.  Effect  of  Gas  on  the  Electron  Emission.  Surface  Effect. 
It  was  shown  in  section  19  that  the  relation  between  the  satura- 
tion thermionic  current  and  the  temperature  of  the  cathode  can 
be  expressed  by  Richardson's  equation: 


where  A  is  a  constant  depending  on  the  number  of  electrons  per 
cubic  centimeter  of  the  cathode,  b  a  measure  of  the  work  which 
an  electron  must  do  to  escape  from  the  cathode,  and  T  the  tem- 
perature of  the  cathode  in  absolute  Kelvin  degrees.  If  the  vacuum 
in  the  tube  is  supposed  to  be  perfect  and  the  electrodes  entirely 
void  of  gas,  the  constants  A  and  b  of  the  above  equation  have 
definite  fixed  values  that  are  determined  only  by  the  nature  of  the 
cathode.  Richardson's  equation  holds  for  any  hot  cathode  and  is 

KAYE  and  LAPY)  it  is  found  to  be  3.5X10"6  cm.  at  atmospheric  pressure. 
This  value  of  the  viscosity  coefficient  is  obtained  by  extrapolation  and  possibly 
involves  a  considerable  error.  It  is  likely  that  the  value  6XlO~6  cm.  for  the 
mean  free  path  of  mercury  vapor  is  more  nearly  correct. 


INFLUENCE  OF  GAS  ON  THE  DISCHARGE  99 

not  dependent  on  the  structural  dimensions  of  the  device,  it  being 
understood  that  the  thermionic  current  Is  is  the  current  obtained 
from  unit  area  of  the  cathode  surface  when  the  potential  difference 
between  cathode  and  anode  is  high  enough  to  draw  all  the  emitted 
electrons  to  the  anode  as  fast  as  they  are  emitted  from  the  cathode. 
It  follows  then  that  if  the  cathode  contains  impurities,  Richard- 
son's equation  must  still  hold  but  the  constants  A  and  b  will  have 
different  values.  From  the  nature  of  the  equation  it  is  seen 
that  while  the  current  changes  in  proportion  to  a  change  in  A, 
a  small  change  in  b  causes  a  very  considerable  change  in  the 
current.  It  has  been  known  for  a  long  time  that  small  traces  of 
gas  occluded  in  the  cathode  can  cause  large  changes  in.  the  ther- 
mionic emission.  H.  A.  Wilson  1  has  found,  for  example,  that 
when  a  platinum  wire  is  freed  of  the  hydrogen  occluded  in  it, 
the  thermionic  current  drops  to  a  very  small  fraction  of  the  value 
obtained  from  a  platinum  wire  not  so  treated.  J.  J.  Thomson  2 
and  O.  W.  Richardson  have  pointed  out  that  the  effect  of  the  gas 
occluded  in  the  surface  of  the  cathode  is  to  change  the  work 
necessary  to  detach  an  electron  from  the  cathode,  that  is,  to 
change  the  constant  b  in  Richardson's  equation  and  so  produce 
relatively  very  large  changes  in  the  thermionic  current.  Thus, 
if  6  =  5X104  (0  =  4.3,  see  equation  (9),  Chapter  III)  and  the 
temperature  of  the  cathode  is  2000°  K.,  an  increase  in  b  of  25 
per  cent  can  decrease  the  current  to  approximately  -^j  of  its 
original  value.  .Such  changes  can  readily  be  produced  by  very 
small  quantities  of  gas  coming  in  contact  with  the  cathode.  The 
amount  of  gas  that  is  necessary  to  produce  great  changes  in  the 
saturation  part  of  the  characteristic  is  often  so  small  that  its 
presence  does  not  noticeably  affect  the  infra-saturation  part  of 
the  characteristic.  This  is  shown,  for  example,  in  Fig.  30.  The 
curves  shown  in  this  figure  were  obtained  with  a  bulb  containing 
two  tungsten  filaments,  one  of  which  was  used  as  cathode  and  the 
other  as  anode.  The  bulb  was  not  baked  during  the  process  of 
evacuation,  so  that  a  small  trace  of  gas  and  water  vapor  remained 
in  the  tube.  It  is  seen  from  curve  1  that  the  pressure  of  the 
residual  gas  was  not  sufficient  to  cause  any  appreciable  increase 
in  the  current  on  the  lower  or  operating  part  of  the  character- 

1  H.  A.  WILSON,  Phil.  Trans.,  Vol.  202,  p.  243,  1903. 

2  J.  J.  THOMSON,  "  Conduction  of  Electricity  through  Gases,"  2d  Ed.,  p. 
202. 


100 


THERMIONIC  VACUUM  TUBE 


istic.  As  the  voltage  and  current  were  increased,  however,  the 
heating  of  the  bulb  by  the  energy  dissipated  in  the  tube  caused 
the  liberation  of  a  sufficient  amount  of  gas  to  give  the  irregular 
curve,  as  evidenced  at  voltages  higher  than  about  200  volts.  The 
readings  were  obtained  in  the  order  indicated  by  the  arrow.  Curve 
2  was  obtained  while  the  whole  tube  remained  immersed  in  liquid 
air.  It  therefore  represents  the  high  vacuum  characteristic. 


Anode  Milliamperes 


FIG.  30 

The  reduction  in  the  electron  emission  caused  by  the  presence 
of  gas  generally  becomes  more  pronounced  when  ionization  takes 
place,  because  this  has  the  effect  of  directing  the  flow  of  gas  towards 
the  filament.  When  the  gas  is  not  ionized  the  electric  field  has  no 
effect  on  the  motion  of  its  molecules,  and  the  chance  of  their 
striking  the  surface  of  the  filament  is  determined  by  he  laws  of 
the  kinetic  theory  of  gases  applicable  at  low  pressures.  When 
ionization  by  collision  takes  place, .  however,  the  molecules  in  the 
space  between  cathode  and  anode  become  positive  ions  which  are 


INFLUENCE  OF  GAS  ON  THE  DISCHARGE  101 

directed  by  the  electric  field  towards  the  cathode  where  they 
recombine  with  the  electrons  into  neutral  gas  molecules.  When 
they  recombine  before  reaching  the  cathode  the  resulting  neutral 
molecules  or  atoms  have  an  increased  momentum  in  the  direction 
of  the  cathode,  due  to  the  momentum  acquired  in  the  electric 
field  while  they  were  ions.  lonization  by  collision  therefore  causes 
more  gas  molecules  to  come  in  contact  with  the  surface  of  the 
cathode.  It  has  been  found,  for  example,  that  the  receiver  type 
of  three-electrode  tubes,  containing  oxide-coated  filaments  and 
that  are  evacuated  sufficiently  well  to  operate  very  satisfactorily 
as  amplifiers,  may  still  contain  a  sufficient  amount  of  gas  to 
paralyze  the  tubes  when  operated  as  oscillation  generators.  (See 
Chapters  VIII  and  IX.)  When  the  tube  is  used  as  an  amplifier 
most  of  the  ionization  of  the  gas  takes  place  between  the  grid 
and  the  anode,  and  since  the  grid  is  negative  with  respect  to  the 
anode,  the  positive  ions  formed  by  collision  ionization  in  this  region 
are  attracted  to  the  grid.  On  the  other  hand,  when  the  tube 
operates  as  an  oscillation  generator,  the  grid  is  subject  to  large 
potential  variations,  so  that  in  this  case  positive  ions  are  also 
formed  in  the  space  between  filament  and  grid.  These  positive 
ions  move  to  the  filament  and  there  combine  with  the  electrons 
to  form  neutral  molecules.  If  the  gas  remaining  in  the  tube  is  of 
such  a  nature  as  to  decrease  the  electron  emission  when  coming 
in  contact  with  the  surface  of  the  filament,  this  effect  can  easily 
become  so  large  that  the  space  current  is  reduced  to  practically 
zero.  This  accounts  for  the  phenomenon  that  has  been  observed 
that  a  tube  will  start  to  oscillate  and  after  a  time,  ranging  from 
a  fraction  of  a  second  to  several  seconds,  the  space  current  will 
drop  to  zero  and  the  tube  become  inoperative.  The  normal 
condition  of  the  tube  can  be  restored  readily  by  heating  the 
filament  to  a  higher  temperature  so  as  to  drive  off  the  gas.  Gen- 
erally it  will  recover  automatically  after  a  period  of  time  depend- 
ing upon  the  temperature  of  the  filament.  This  period  may  range 
from  a  fraction  of  a  second  to  several  seconds  or  even  minutes. 
The  best  way  to  prevent  this  paralysis  of  the  tube  is  to  evacuate 
it  more  thoroughly. 

Langmuir  l  has  made  extensive  investigations  on  the  effects 
of  gas  on  electron  emission  from  tungsten  filaments.     Langmuir 

1 1.  LANGMUIR,  Phys.  Rev.,  Vol.  2,  p.  450,  1913;   Phys.  Zeitschr.,  Vol.  15, 
p.  516,  1914 


102  THERMIONIC  VACUUM  TUBE 

finds  that  argon,  mercury  vapor  and  dry  hydrogen  have  no  direct 
effect  on  the  emission  of  electrons  from  tungsten,  but  water  vapor 
has  a  very  large  effect. 

Pure  dry  nitrogen  has  been  found  by  Langmuir  to  have  no 
appreciable  direct  effect  in  reducing  the  electron  emission  when 
the  amount  of  nitrogen  left  in  the  tube  is  so  small  that  there  is  no 
appreciable  ionization  of  the  nitrogen  molecules.  But  when  the 
voltage  is  raised  so  high  that  ionization  becomes  appreciable  the 
nitrogen  ions  can  bombard  the  filament  with  sufficient  velocity 
to  combine  with  the  tungsten.  This  causes  a  decrease  in  the 
electron  emission  from  the  tungsten.  The  higher  the  velocity  with 
which  the  nitrogen  ions  strike  the  tungsten  filament  the  greater 
seems  to  be  the  effect  on  the  electron  emission,  so  that  in  the 
presence  of  nitrogen  the  current  at  first  increases  in  the  manner 
shown  in  the  characteristics  of  most  thermionic  devices,  and  then 
suddenly  starts  to  decrease  when  the  voltage  is  still  further 
increased.  Hence,  instead  of  getting  a  curve  which,  for  voltages 
higher  than  the  saturation  voltage,  becomes  substantially  parallel 
to  the  voltage  axis,  the  curve  obtained  at  these  voltages  has  a 
negative  slope. 

40.  Influence  of  Occluded  Gases.  From  the  explanations 
given  in  Section  38  it  follows  that  the  influence  of  ionization  by 
collision  of  the  residual  gases  in  a  tube  on  the  infra-saturation  part 
of  the  characteristic  is  generally  not  disturbing  for  pressures 
lower  than  of  the  order  of  one-tenth  to  one  micron.  Such  a  pres- 
sure is  easily  obtained.  Hence,  as  far  as  removing  the  gas  in  the 
space  of  the  tube  is  concerned,  there  would  be  no  difficulty  in 
obtaining  a  sufficiently  high  vacuum  to  realize  what  may  be  called 
a  "  pure  electron  discharge."  What  is  necessary,  however,  is  to 
maintain  the  vacuum  in  the  tube  while  it  is  in  operation,  and  it  is 
therefore  not  necessary  merely  to  remove  the  gas  from  the  volume 
of  the  tube,  but  also  to  free  the  electrodes  and  walls  of  the  vessel  of 
occluded  gases  to  a  sufficient  extent.  If  the  electrodes  of  a  device 
remain  cold  during  operation  the  occluded  gases  are  not  liberated 
very  readily,  but  when  the  electrodes  become  hot  during  the  opera- 
tion the  occluded  gases  are  liberated.  In  all  vacuum  devices 
using  hot  electrodes  it  is  therefore  necessary  previously  to  free  the 
electrodes  of  gases.  An  incandescent  lamp  is  such  a  device  and 
therefore  it  has  always  been  common  practice,  in  evacuating 
incandescent  lamps,  to  heat  the  bulbs  and  raise  the  filaments  to 


INFLUENCE  OF  GAS  ON  THE  DISCHARGE  103 

abnormally  high  incandescence  during  the  evacuating  process. 
In  thermionic  devices  usually  only  the  filament  is  such  that 
it  can  be  heated  during  evacuation  by  passing  a  current  through 
it.  The  other  electrodes  are  heated  by  applying  a  positive  poten- 
tial to  them  which  is  sufficiently  high  to  enable  the  electrodes  to 
rise  to  high  temperatures  by  the  bombardment,  of  the  electrons 
coming  from  the  hot  filament. 

The  extent  to  which  the  electrodes  and  walls  of  the  vessel  must 
be  freed  of  gas  depends  on  the  temperature  to  which  these  parts 
of  the  tube  rise  during  operation.  If,  for  example,  a  tube  is 
designed  to  operate  on  voltages  ranging,  say,  from  15  to  50  volts, 
such  as  is  the  case  with  the  type  of  tube  commonly  used  as  detector 
and  amplifier  in  radio-receiving  stations,  it  is  not  necessary  to 
evacuate  the  tube  so  well  that  it  can  also  operate  satisfactorily 
at  much  higher  "voltages.  Such  a  tube  while  operating  satis- 
factorily as  a  pure  electron  discharge  device  over  the  operating 
voltages  stated  above,  may  undergo  a  sufficient  liberation  of 
gas  from  electrodes  to  spoil  the  tube  when  the  voltage  is  raised  to, 
say,  iOO  volts  or  more.  Tubes  that  are  to  operate  on  higher  volt- 
ages and  currents  must  have  their  electrodes  more  thoroughly 
freed  of  gas  during  the  process  of  evacuation. 

The  way  in  which  the  characteristic  is  influenced  by  the  liber- 
ation of  gas  when  a  tube  is  subjected  to  voltages  higher  than  those 
for  which  it  is  designed  is  shown  in  Fig.  31.  These  curves  were 
obtained  with  a  standard  Western  Electric  VTl  tube.  It  contains 
an  oxide-coated  filament  and  is  designed  to  operate  on  voltages 
not  higher  than  100  volts.  If  the  voltage  is  raised  much  beyond 
this  value  the  electrodes  become  hot  enough  to  liberate  some  of  the 
gas  occluded  in  them,  and  the  amount  of  gas  liberated  increases 
as  the  potential  difference  between  filament  and  plate  is  raised. 
This  increases  the  amount  of  ionization  by  collision  and  causes 
the  filament  to  be  bombarded  by  the  positive  ions.  The  bom- 
bardment of  the  filament  raises  its  temperature  and  increases  the 
space  current  over  the  value  that  it  would  have  if  there  were  no 
positive  ion  bombardment.  When  the  voltage  becomes  high 
enough  the  current  increases  rapidly.  Such  a  rapid  increase  in 
the  current  is  shown  in  the  case  of  curve  1  (Fig.  31)  to  take  place 
at  about  400  volts,  and  in  the  case  of  curve  2  at  about  300  volts. 
Tungsten  filaments  do  not  seem  to  be  so  sensitive  to  positive 
ion  bombardment.  It  sometimes  happens  that  the  gas  liberated 


104 


THERMIONIC  VACUUM  TUBE 


from  the  electrodes  has  a  very  pronounced  effect  in  reducing  the 
electron  emission  from  the  cathode  and  then  the  current  instead 
of  increasing  may  decrease  at  the  higher  voltages. 

If  the  voltage  is  raised  to  an  excessive  amount  so  much  gas 
may  be  liberated  as  to  spoil  the  tube  permanently.  On  the  other 
hand,  the  gas  liberated  can  be  cleaned  up  by  the  hot  filament 
so  that  the  tube  automatically  restores  itself.  This  is  especially 
the  case  with  tungsten  filaments. 


100. 


150 


ZOO          Z50          300 
Anode  Vq|ts 

FIG.  31. 


350         400 


450 


50ft 


If  the  amount  of  gas  liberated  by  applying  an  over-voltage  is 
not  excessive  the  tube  will  behave  like  a  high  vacuum  tube  on  the 
lower  or  operating  part  of  the  characteristic,  even  after  the  gas 
has  been  liberated  at  the  voltages  corresponding  to  the  saturation 
part.  This  is  shown,  for  example,  by  Fig.  32,  which  represents 
a  curve  also  obtained  with  a  VT1  tube.  When  the  voltage  was 
raised  to  about  250  or  300  volts,  the  current  began  to  increase 
rapidly,  as  shown  by  the  continuous  line.  The  broken  line 


INFLUENCE  OF  GAS  ON  THE  DISCHARGE  105 

shows  the  currents  obtained  for  decreasing  voltage  after  the  voltage 
had  been  raised  to  about  340  volts.  It  will  be  seen  that  on  the 
upper  part  of  the  characteristic  the  currents  obtained  for  de- 
creasing voltages  differ  considerably  from  those  obtained  for 
increasing  voltages.  On  the  infra-saturation  part  of  the  charac- 
teristic, however,  the  two  curves  coincide  very  well,  showing  that 
the  amount  of  gas  in  the  tube  is  not  sufficient  to  cause  any  appre- 
ciable deviation  for  voltages  lower  than  about  100  volts. 

Thermionic  tubes  as  they  are  used  to-day  perform  a  large 
number  of  different  functions,  most  of  which  require  that  the  char- 
acteristic be  steady  and  reproducible.  This  is,  for  example,  the 


10 


,150  ZOO 

Anode  Volts 

FIG.  32. 


case  where  the  tube  is  used  as  a  telephone  repeater.  In  order  to 
insure  this  the  procedure  commonly  adopted  in  practice  is  to 
apply  a  potential  for  about  a  minute  or  so,  to  the  plate,  which 
is  higher  than  the  normal  operating  voltages,  and  then  test  the 
tube  to  see  if  it  performs  properly  the  function  for  which  it  is 
designed.  This  test  is  commonly  referred  to  as  the  "  over- 
voltage  test."  If  the  tube  is  not  sufficiently  well  evacuated  the 
gas  is  liberated  from  the  electrodes  during  the  time  that  the  over- 
voltage  is  applied.  If  the  tube  still  functions  properly  after  the 
application  of  the  over-voltage,  it  means  that  the  amount  of  gas 
liberated  was  not  sufficient  to  have  any  deleterious  effect  on  the 
operating  part  of  the  characteristic.  The  difference  between  the 


106  THERMIONIC  VACUUM  TUBE 

over- volt  age  to  be  used  in  this  test  and  the  highest  normal  operating 
voltage  depends  on  the  margin  of  safety  that  it  is  desired  to  secure. 
41.  lonization  at  High  Pressures.     The  ionization  phenomena 
encountered  in  thermionic  tubes  belong  to  a  class  where  the  mean 
free  path  of  the  electrons  in  the  gas  is  generally  great  compared 
with    the    distance    between    the    electrodes.     The    phenomena 
resulting  from  the  discharge  at  such  pressures  that  the  mean 
free  path  is  smaller  than  the  distance  between  the  electrodes  are 
more  complicated  and  show  the  effect  of  cumulative  ionization. 
In  order  to  show  this  in  an  elementary  way  let  us 
consider  what  happens  when  the  mean  free  path  is 
so  small  that  in  passing  from  cathode  to  anode  an 
electron  has  a  chance  of  colliding  several  times  with 
gas  molecules.     Let  the  number  of  electrons  start- 
ing from  the  cathode  be  no,  and  let  the  number  of 
electrons  formed  by  collision  ionization  in  travers- 
ing a  distance  x  from  the  cathode  be  n.      (See  Fig. 
33.)     The  total  number  of  electrons  arriving  at  a 
plane  distant  x  from  the  cathode  is  then  n+no. 
FIG.  33.         If  each  electron  in  moving  through  unit  distance 
can    dislodge  a    other  electrons    by  collision,  the 
number  dislodged   in   a   region   of   thickness  dx  at  a  distance  x 
from  the  cathode  will  be* 

dn=(no+ri)adx. 

To  find  the  total  number  N  arriving  at  the  anode,  we  have  to  in- 
tegrate this  equation  between  the  limits  of  n  =  0  when  x  =  Q, 
and  no~i-n  =  N  when  x  —  d,  the  distance  between  cathode  and 
anode.  This  gives: 

N  =  n0ead. 

This  equation  shows  that  the  number  of  electrons  reaching  the 
anode,  and  therefore  also  the  current  through  the  tube,  increases 
with  increasing  distance  d  between  cathode  and  anode.  This 
is  in  marked  contrast  to  the  discharge  in  high  vacuum  thermionic 
tubes  in  which,  as  was  shown  in  the  previous  chapter,  the  satura- 
tion current  is  independent  of  the  distance  between  cathode  and 
anode,  while  the  infra-saturation  current  decreases  as  the  distance 
between  cathode  and  anode  is  increased. 


INFLUENCE  OF  GAS  ON  THE  DISCHARGE  107 

42.  Difference  between  Gas-free  Discharge  and  Arc  Discharge. 

There  are  other  important  differences  between  these  two  different 
types  of  discharge.  The  mercury  arc  is  an  example  of  a  practical 
device  which  depends  for  its  operation  on  ionization  by  collision, 
the  gaseous  medium  being  mercury  vapor  in  equilibrium  with  the 
liquid  mercury  used  as  cathode.  In  a  device  containing  a  consider- 
able amount  of  gas  and  cold  electrodes  the  discharge  will  not  pass 
unless  the  applied  voltage  be  made  so  high  that  the  few  electrons 
in  the  space  can  cause  a  small  initial  ionization.  The  positive 
ions  so  formed  bombard  the  cathode  and  give  up  sufficient  energy 
to  the  cathode  to  enable  the  electrons  to  overcome  the  force  of 
attraction  at  the  surface  of  the  cathode  and  so  escape  from  it. 
The  discharge  may  also  be  started,  as  is  done  in  the  mercury  arc, 
by  bringing  the  cathode  in  contact  with  an  auxiliary  electrode 
and  then  striking  the  arc  by  separating  them.  This  furnishes 
the  initial  ionization  necessary  to  start  the  discharge.  Thus, 
while  electrons  are  liberated  from  the  cathode  in  the  pure  electron 
device  simply  by  external  heating  cf  the  cathode  such  as  passing 
a  heating  current  through  it,  in  the  gas-filled  tube  the  electrons 
are  liberated  by  bombardment  of  positive  ions  and  also  by  photo- 
electric effects  in  the  tube. 

The  positive  ions  formed  by  collision  ionization  move  toward 
the  cathode  and  the  electrons  toward  the  anode.  There  is,  thus, 
a  predominance  of  positive  space  charge  in  the  neighborhood 
of  the  cathode  and  a  predominance  of  negative  space  charge  near 
the  anode.  When  the  conditions  are  such  that  an  arc  discharge 
passes,  the  total  space  charge  is  small  compared  with  that  in  the 
gas-free  tube,  where  the  space  charge  is  negative  only  and  has  a 
maximum  value  near  the  cathode.  The  resistance  of  the  arc  is 
therefore  lower  than  that  of  a  gas-free  tube.  In  order  to  maintain 
an  arc  steady  it  is  necessary  to  connect  it  in  series  with  an  external 
resistance.  The  gas-free  tube,  on  the  other  hand,  does  not  need 
an  external  resistance  to  stabilize  the  discharge.  On  account  of 
the  frequent  collisions  in  an  arc  there  is  also  a  great  deal  of  recom- 
bination and  this  causes  a  pronounced  blue  glow  in  the  tube. 
The  gas-free  tube,  on  the  other  hand,  shows  no  blue  glow.  If  a 
blue  glow  does  accidentally  appear,  it  is  because  the  tube  has  been 
over-taxed  and  it  may  cause  the  tube  to  become  inoperative. 

Another  important  difference  between  a  pure  electron  discharge 
and  an  arc  discharge  is  that  the  latter  has  a  "  falling  characteris- 


108 


THERMIONIC  VACUUM  TUBE 


tic";  that  is,  its  relation  between  current  and  voltage  is  given 
by  a  Curve  such  as  AB  (Fig.  34).  The  gas-free  device,  on  the  other 
hand,  has  a  characteristic  similar  to  OC.  The  difference  between 
tubes  containing  these  characteristics  becomes  apparent  when  we 
consider  the  corresponding  a-c.  resistances.  The  a-c.  resist- 
ance for  small  current  or  voltage  variations  at  any  definite  voltage 

is  given  by  the  reciprocal  of 
the  slope  of  the  characteristic 
at  a  point  corresponding  to 
that  voltage.  Since  the  slope 
of  the  curve  AB  is  negative, 
the  arc  has  a  negative  resist- 
ance, while  the  resistance  of 
a  gas-free  tube  is  positive. 

It    is  the  negative   resist- 
ance of  the  arc  which  enables 
it  to  produce  sustained  oscil- 
lations.    It  will  be  shown  in 
Chapter  VIII  that    a    device 


Voltage 

FIG.  34. 

containing  only  two  electrodes 

can  only  produce  sustained  oscillations  if  it  has  a  negative 
resistance  or  a  falling  characteristic.  The  principle  involved  in 
the  production  of  sustained  oscillations  by  the  audion  or  three- 
electrode  thermionic  tube  is  entirely  different  and  depends  on  the 
controlling  action  of  the  grid  on  the  electron  flow  from  filament 
to  anode. 


CHAPTER  VI 

RECTIFICATION   OF    CURRENTS   BY  THE  THERMIONIC 

VALVE 

43.  Conditions  for  Rectification.  Let  us  consider  a  device 
on  which  can  be  impressed  a  simple  harmonic  voltage  and  let  the 
current  through  the  device  be  any  function  f(e)  of  the  voltage, 
This  function  can  always  be  expressed  in  a  Fourier  series,  thus: 

n  n 

f(e)=lQ+2ansmnx+2bncosnx,     .     .     .     .     (1) 
i  i 

where  /o,  «n  and  bn  are  constants. 

The  summation  terms  of  this  series  are  simple  harmonic 
functions,  and  will  therefore  vanish  when  integrated  over  a  com- 
plete period.  On  the  other  hand  IQ,  being  a  constant,  will  be 
independent  of  such  integration  and  can  be  measured  with  a  d.  c. 
measuring  instrument.  Hence,  provided  that  IQ  be  not  zero, 
the  function  f(e)  will  be  such  that  the  device  will  rectify.  The 
fundamental  condition  for  rectification  by  any  device  is  therefore: 


r 


.     .     (2) 

This  will  be  the  case  either  when 

rf(e)dt  =  Q    or      /  *  f(e)dt=0     ....     (3) 
or  when 


*  C 

JT 


Tf(e)dt (4) 


Devices  which  comply  with  condition  (3)  are:    (1)  those  which 
conduct  current  only  in  one  direction  and  for  which  f(e)  may  be 

109 


110 


THERMIONIC  VACUUM  TUBE 


any  function  of  e  in  the  transmission  half  period  as,  for  example, 
the  thermionic  rectifier  (Fig.  35);  (2)  those  which  conduct 
current  only  in  one  direction  and  for  which  f(e)  is  any  finite 
function  of  e  for  all  values  of  e  greater  than  a  minimum  value  e\. 
The  electrolytic  rectifier  practically  complies  with  this  condition; 


-  e  + 


FIG.  35. 


during  the  transmission  half  period  it  does  not  conduct  unless  the 
applied  voltage  exceeds  its  back  E.M.F.     (Fig.  36.) 

Devices  which  comply  with  condition  (4)  are:  (a)  those 
which  conduct  current  in  both  directions  but  for  which  f(e)  is 
unsymmetrical  with  respect  to  the  axis  of  current  (Fig.  37); 
(6)  those  for  which  f(e)  is  a  linear  function  of  e,  provided  the 
input  voltage  exceeds  a  minimum  value  e\  (Fig.  38). 


FIG.  36. 

The  three-electrode  thermionic  detector,  or  audion,  cannot 
be  called  a  rectifier,  as  far  as  the  plate  current  is  concerned,  because 
it  does  not  rectify  the  incoming  current.  This  current  only 
releases  energy  in  the  plate  circuit  which  is  supplied  by  the  local 
plate  battery,  and  the  characteristic  of  the  device  is  such  that  more 
energy  is  released  during  the  one-half  period  than  during  the  other. 
It  will  be  seen  that  devices  represented  by  the  conditions  (3)  can 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE    111 


be  made  to  comply  with  condition  (4)  by  inserting  a  local  battery 
in  the  rectifier  circuit  so  a?  to  shift  the  axis  of  current.  But 
even  with  this  modification  they  differ  from  the  audion  detector 
because  they  actually  resolve  the  incoming  currents  into  d-c. 


e 
FIG.  37. 

and  a-c.  components  in  the  measuring  circuit.  The  three- 
electrode  device  or  audion  therefore  differs  radically  from  these 
other  types  of  radio  detectors. 

A  full  discussion  of  the  operation  of  the  various  types  of 
rectifiers  is  beyond  the  scope  of  these  pages.  We  shall  therefore 
confine  our  attention  to  the  thermionic  rectifier  or  valve. 


Tel.  Rec. 
FIG.  39. 

44.  The  Fleming  Valve.  This  device  satisfies  condition  (3) 
and  has  a  characteristic  such  as  that  shown  in  Fig.  35  (6).  It 
consists  of  a  filament  which  can  be  heated  to  incandescence  and  a 
plate,  both  placed  in  a  highly  evacuated  bulb.  In  1G05  Fleming  l 
recognized  the  use  of  the  rectifying  properties  of  this  device  for 
the  indication  of  high  frequency  oscillations,  and  used  it  as  a 

1  J.  A.  FLEMING,  Proc.  Roy.  Soc.,  Jan.,  1905,  p.  476;  IL  S.  Pat.,  803,  684. 


112 


THERMIONIC  VACUUM  TUBE 


radio  detector.     The  circuit  in  which  Marconi  used  this  device 
as  a  radio  detector  is  shown  in  Fig.  39. 

45.  Valve  Detector  with  Auxiliary  Anode  Battery.  By  our 
present  standard  of  measurement  the  two-electrode  tube  is  a 
very  inefficient  detector.  It  can  be,  and  has  been  used  more 
efficiently  by  operating  on  a  chosen  point  of  the  current-voltage 
characteristic,  thus  making  it  fall  in  the  class  of  rectifiers  given  by 
condition  (4)  instead  of  that  represented  by  condition  (3).  This 
can  be  done  by  inserting  a  local  battery  in  the  circuit  as  shown 
in  Fig.  40.1 


Tel.  foe. 

FIG.  40. 

The  operation  of  the  device  when  used  this  way  can  be  under- 
stood from  the  following :  By  the  insertion  of  the  battery  E  there 
is  established  in  the  circuit  FPE  a  constant  direct  current  which 
has  a  finite  value  even  when  no  oscillations  are  impressed  from  the 
antenna.  The  current  through  the  device  can  therefore  be  repre- 
sented by  a  function  of  the  form 


I+i=f(E+esmpt), 


(5) 


where  E  is  the  local  source  of  direct  voltage,  I  the  direct  current 
due  to  E  and  i  the  superposed  a-c.  due  to  e.  This  can  be  expanded 
into  a  power  series: 

f(E+e  sin  pt)  =f(E)+f'(E)e  sin  pt 


e2  sin2  pt 


+ 


en  sinn  pi 


!See  LEE  DE  FOREST,  Proc.  A.I.E.E.,  Vol.  25,  p.  719,  1906,  and  J.  A. 
FLEMING,  Proc.  Roy.  Inst.,  Great  Britain,  June,  1909,  p.  677. 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE     113 
that  is 

r"(E)^+  .  .  .  +f(E)esmpt 

f/ffm^cos2pt 
-f  (E) ~A +  ...     (6) 


If  now  this  be  integrated  over  a  complete  period  the  sine 
and  cosine  terms  vanish.  Of  the  remainder  the  termf(E)  represents 
simply  the  direct  current  established  in  the  circuit  FPE  by  the 

e2 
battery  E}  and  the  series  f"(E)--\-  .  .  .  represents  direct  current 

component  established  by  the  incoming  oscillations.  This  series 
represents  second  and  higher  derivatives  of  the  characteristic 
and  hence  if  /  is  a  linear  function,  the  series  vanishes  and  the 
device  does  not  rectify.  But  the  characteristic  of  the  thermionic 
valve  is  not  linear;  the  derivative  series  is  therefore  finite.  This 
series  generally  converges  so  rapidly  that  all  the  terms  except  the 

e2 
first,  }"(E)—,  can  be  neglected,  so  that  the  rectified  current  can 

be  given  by  the  second  derivative  of  the  characteristic.  It  also 
follows  from  this  that  the  rectified  current  is  proportional  to  the 
square  of  the  input  voltage  e. 

As  an  example  let  us  take  an  arbitrary  case  in  which  the 
current  in  the  device  is  proportional  to  the  nth  power  of  the 
voltage,  thus:  I  =  aEn.  The  rectified  current  is  then  given  by 

~-n(n 

If,  for  example,  the  current  varies  as  the  f-power  of  the  applied 
voltage  (n  =  %)  the  rectified  current  is  inversely  proportional  to 
the  square  root  of  the  locally  applied  voltage.  If  n  =  2  the  rectified 
current  is  independent  of  the  local  voltage  while  for  higher  values 
of  n,  it  increases  with  the  local  voltage. 

It  is  well  known  that  none  of  these  cases  applies  to  the  valve 
when  used  as  a  radio  detector,  but  that  the  rectified  current  shows 
a  maximum  for  a  definite  value  of  the  local  voltage.  This  is  due 
to  the  fact  that  the  cathode  is  not  an  equipotential  surface  but  a 
filament  in  which  is  established  a  voltage  drop  due  to  the  heating 
current. 

In   Chapter  IV  it  was  shown  that  if   the  voltage   drop  in 


114 


THERMIONIC  VACUUM  TUBE 


the  filament  is  taken  into  account  the  characteristic  of  the  valve 
can  be  represented  by  the  two  following  equations: 


}E<Ej 


(17') 


for  all  values  of  E  less  than  the  voltage  drop  in  the  filament,  Ef, 
and 

....     (180 


for  values  of  E  greater  than  Ef. 

If  /  be  computed  from  these  equations  for  arbitrary  values  of 
E  within  the  respective  limits,  the  two  curves  obtained  will  fit 
to  form  a  continuous  curve  such  as  the  curve  OA  in  Fig.  17  (p.  51). 
But  this  is  not  the  case  with  the  relation  between  the  voltage 
and  the  second  derivative  of  the  current.  The  second  derivatives 
of  the  above  equations  are 

=  A#V2  (7) 

TT<^          rr -^  \'/ 


Anode  VoU-s 
FIG.  41. 


If  these  expressions  be  plotted  for  arbitrary  values  of  E,  the 
result  is  a  curve  which  shows  a  distinct  maximum  at  a  value 
of  the  local  voltage  E  equal  to  the  voltage  drop  1  in  the  filament, 
(Fig.  41).  This  therefore  accounts  for  the  observed  maximum 


1  It  is  to  be  understood  that  this  voltage  E  is  the  sum  of  the  voltage  applied 
and  the  contact  potential  difference  between  filament  and  anode. 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE     115 

in  rectified  current  obtained  at  a  suitably  adjusted  voltage  of  the 
battery  in  the  valve  circuit. 

But  even  with  this  adjustment  the  valve  falls  far  short  of  the 
three-electrode  device  which  has  now  completely  superseded  it 
as  a  radio  detector.  The  discussion  of  the  valve  as  a  radio  detec- 
tor will  therefore  be  limited  to  the  elementary  theoretical  considera- 
tions given  above,  which  are  generally  applicable  and  can  with  some 
modification  be  adapted  to  the  treatment  of  the  three-electrode 
device. 

46.  Thermionic  Valve  as  High  Power  Rectifier.  The  ther- 
mionic valve  has  considerable  practical  value  as  a  rectifier  of 


currents  for  high-power  purposes,  and  has  been  used  successfully 
for  the  production  of  unidirectional  currents  at  high- volt  ages.  By 
a  proper  arrangement-  of  circuits  it  can  also  be  used  to  convert 
alternating  current  into  steady  direct  current.  When  this  device 
is  used  for  rectifying  currents  at  high  voltages  it  does  not  mean 
that  it  must  be  so  constructed  as  to  transmit  current  with  these 
high  voltages  at  its  terminals.  As  a  matter  of  fact,  even  in  cases 
where  the  voltage  to  be  rectified  is  as  high  as  100,000  volts  and 
more,  the  voltage  drop  in  the  valve  when  it  transmits  current 
is  only  a  few  hundred  volts  and  sometimes  much  less  and  the 
power  dissipated  in  the  valve  is,  comparatively  speaking,  very 
small. 

In  order  to  explain  the  operation  of  the  thermionic  valve  in  a 
rectifier  circuit,  let  us  consider  its  d-c.  characteristic  curve,  shown 
in  Fig.  42.  When  operated  under  the  right  conditions  the  device 


116 


THERMIONIC  VACUUM  TUBE 


conducts  practically  no  current  during  the  half  cycle  when  the 
filament  is  positive  with  respect  to  the  anode.  During  the  other 
half  cycle  a  current  wave  shape  is  obtained  depending  on  the 
value  of  the  applied  voltage  and  upon  the  shape  of  the  d-c.  char- 


FIG.  43. 

acteristic  of  the  valve.  As  long  as  the  peak  value  of  the  applied 
simple  harmonic  voltage  is  less  than  Es  (Fig.  42)  when  the  fila- 
ment temperature  is  T  the  current  wave  shape  takes  the  form 
shown  by  the  continuous  curves  in  Fig.  43,  the  applied  voltage 
being  given  by  the  broken  curve  in  arbitrary  scale.  The  departure 


FIG.  44. 

of  the  current  curve  from  the  simple  harmonic  shape  is  due  to  the 
non-linear  current-voltage  characteristic  of  the  valve. 

If  the  voltage  applied  between  the  terminals  of  the  valve 
exceeds  the  value  given  by  Es  (Fig.  42)  the  current  curve  will  be 
flattened  as  shown  in  Fig.  44. 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE     117 

47.  Optimum  Voltage  for  Rectification.  An  increase  of  the 
voltage  beyond  Es  causes  no  further  increase  in  current  and  the 
valve  then  obviously  operates  inefficiently.  Furthermore,  if  the 
applied  voltage  is  less  than  Es  the  valve  also  operates  inefficiently; 
because  in  this  case  more  power  is  expanded  in  heating  the  filament 
than  is  necessary.  This  can  be  understood  by  remembering  that 
the  saturation  current  increases  with  the  temperature.  It  is  seen, 
therefore,  that  the  valve  operates  most  efficiently  when  the  maxi- 
mum value  of  the  voltage  applied  between  its  terminals  is  equal 
to  the  d.c.  voltage  that  is  just  sufficient  to  produce  the  saturation 
current.  How  this  voltage  depends  on  the  constants  of  the 
tube  can  be  seen  from  the  following  consideration.  The  satura- 

-- 
tion  is  given  by  Richardson's  equation,  Is  =  AcATl/2e   T,  where 

Ac  is  the  area  of  the  cathode,  T  the  temperature  of  the  cathode  and 
A  and  b  constants  the  meaning  of  which  was  explained  in  Section 
19.  This  equation  holds  for  the  flat  portion  A  B  of  the  char- 
acteristic (see  Fig.  42). 

As  was  explained  in  Chapter  IV  the  current  on  the  lower  part 
of  the  characteristic  is  limited  by  the  space  charge,  the  voltage 
drop  in  the  filament,  etc.,  'so  that  in  general  the  current  cannot  be 
taken  to  vary  as  the  f-power  of  the  voltage  between  filament 
and  anode.  The  limitation  of  the  current  by  the  voltage  drop 
in  the  filament,  for  example,  causes  the  current  to  be  smaller  than  in 
the  theoretical  case  of  an  quipotential  cathode,  but  to  increase 
at  a  greater  rate  than  the  f-power  of  the  voltage.  We  shall 
therefore  assume  that  the  current  on  the  infra-saturation  part 
of  the  characteristic  is  proportional  to  the  nth  power  of  the  applied 
voltage. 

Considering  that  we  are  interested  in  determining  the  optimum 
voltage  which  occurs  at  the  knee  of  the  characteristic,  it  must 
be  noted  that  the  limitation  of  current  by  electron  emission  from 
the  filament,  which  usually  extends  into  the  lower  part  of  the 
characteristic  and  is  not  confined  only  to  the  saturation  region, 
causes  the  characteristic  to  bend  over  to  the  right,  so  that  it  is 
not  possible  to  express  the  whole  infra-saturation  part  of  the  char- 
acteristic by  a  simple  power  relation  between  current  and  voltage. 
We  can,  however,  in  the  present  consideration  neglect  this  effect 
and  determine  the  point  A  at  the  intersection  of  the  saturation 
current  and  the  current  given  by  I  =  CEn.  This  would  correspond 
sufficiently  closely  to  the  optimum  voltage. 


118  THERMIONIC  VACUUM  TUBE 

The  current  below  A  will  furthermore  depend  on  the  areas 
of  the  anode  and  cathode  and  the  distance  between  them.     Hence: 

I=f(Ae,Aa,  d)En.          .    .     .    .     .     (9) 

where    Ac  =  area  of  cathode  ; 
A0  =  area  of  anode; 
d  =  distance  between  cathode  and  anode. 

At  the  point  A  the  characteristic  is  obeyed  by  both  equations 
//  =  /5.     Hence: 


c 

f(Ac,Aa,dy  ;  ; 

This  gives  the  voltage  that  should  be  applied  to  the  terminals 
of  the  rectifier  to  make  it  operate  most  efficiently. 

Instead  of  using  Richardson's  equation  for  the  saturation 
current  we  can  make  use  of  the  simpler  equation  given  on  p.  78 
which  holds  with  sufficient  accuracy  over  the  whole  range  of  tem- 
peratures that  one  might  want  to  use  in  practice.  This  equation  is 

I=AjCp»+1,       .    .    w,    .    .     (11) 

where  /  is  the  current  from  a  cathode  of  area  Ae,  p  the  power 
dissipated  in  heating  the  cathode  and  n  an  empirically  determined 
exponent. 

The  voltage  Es  can  then  be  obtained  by  eliminating  /  from 
this  equation  and  equation  (9).  Before  doing  so  let  us  obtain  an 
expression  for  the  function  /  in  equation  (9).  This  equation  was 
written  in  the  functional  form  to  apply  it  to  the  case  in  which  both 
filament  and  anode  are  of  finite  size.  For  such  a  tube  in  which 
the  anode  is  in  the  form  of  a  plate  or  plates  the  current  is  practi- 
cally proportional  to  the  area  of  both  anode  and  cathode  and 
inversely  proportional  to  the  square  of  the  distance  between  them. 
Equation  (9)  can  therefore  be  written  approximately: 


(12) 


Hence  from  (11)  and  (12)  we  obtain 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE    119 

where  Es  is  the  value  of  E  corresponding  to  the  point  where  the 
characteristic  bends  over.  The  constant  C  depends  upon  the  ther- 
mionic efficiency  of  the  filament  (see  equation  (24),  Chapter  IV). 
For  example,  for  tungsten  C  has  the  value  1.8X  10~7  and  for  a  type 
of  Western  Electric  filament  the  value  2.15X10"2. 

It  is  therefore  seen  that  the  optimum  voltage  drop  in  the 
rectifier  depends  upon  the  thermionic  efficiency  of  the  cathode, 
the  power  per  cm.2  used  for  heating  the  cathode,  the  area  of  the 
anode  and  the  distance  between  cathode  and  anode. 

The  minimum  voltage  Es  necessary  to  obtain  the  saturation 
current  is  an  important  quantity,  not  only  in  dealing  with  the 


rectifier  but  also  in  designing  the  three-electrode  type  of  ther- 
mionic tube.  We  shall  therefore  make  the  above  relationship 
somewhat  clearer  by  considering  the  current-voltage  character- 
istic. Let  us  start  with  a  tube  having  certain  values  for  the 
parameters  appearing  in  equation  (13),  and  let  the  current- 
voltage  characteristic  of  this  tube  be  represented  in  arbitrary 
scale  by  OAB  (Fig.  45).  The  voltage  Es  of  equation  (13)  is  given 
by  the  projection  of  the  point  A  on  the  voltage  axis.  Now  let 
the  size  of  the  anode  be  increased  to  have  double  the  area.  The 
infra-saturation  current  (given  by  the  curved  part  of  the  char- 
acteristic) will  be  doubled,  but  the  saturation  current  (given  by 
the  horizontal  portion)  is  independent  of  the  area  of  the  anode. 
The  characteristic  will  therefore  take  the  form  OCB  and  the 


120  THERMIONIC  VACUUM  TUBE 

optimum  voltage  Es  will  be  less,  although  the  maximum  current 
is  the  same  as  before.  The  result  is  therefore  a  better  rectifier. 
Now,  suppose  the  distance  d  between  cathode  and  anode  be 
increased.  This  will  not  change  the  saturation  but  will  reduce 
the  infra-saturation  current,  and  the  characteristic  may  be  repre- 
sented by  a  curve  such  as  ODB.  This  increases  the  voltage  Es. 
If  the  anode  and  cathode  be  kept  as  initially  but  the  power  dis- 
sipated per  cm.2  at  the  cathode,  i.e.,  its  temperature,  be  increased 
the  characteristic  takes  the  form  OEG.  This  increases  the  voltage 
Es  but  at  the  same  time  the  total  current  is  increased.  Further- 
more, an  increase  in  the  thermionic  efficiency  of  the  cathode 
produces  a  similar  effect  to  that  resulting  from  an  increase  in  the 
temperature,  so  that  an  appropriate  increase  in  the  thermionic 
efficiency  would  also  change  the  characteristic  to  the  curve 
OEG. 

It  will  be  noticed  that  the  area  of  the  cathode  does  not  enter 
into  equation  (13),  which  means  that  the  optimum  voltage  is 
independent  of  it.  This  can  easily  be  understood  if  we  consider 
that  a  change  in  the  cathode  area,  say  by  changing  the  length  of 
the  filament,  produces  an  equal  change  in  both  the  space  charge 
and  saturation  currents.  Thus,  if  the  filament  length  be  doubled 
the  characteristic  obtained  will  be  OFH  instead  of  OAB)  and  it 
is  seen  that  the  voltage  Es  has  the  same  value  as  before,  namely 
that  corresponding  to  the  point  A  or  F,  but  the  total  current  will 
be  increased. 

48  Types  of  Thermionic  Valves.  In  designing  a  valve  to 
operate  on  very  high  voltages  certain  important  factors  must  be 
taken  into  consideration.  In  the  first  place  the  high  potential 
difference  existing  between  the  electrodes  during  part  of  the 
blocking  half  period  causes  a  mechanical  strain  which  tends  to 
pull  the  filament  over  to  the  anode,  and  this  force  will  be  the 
greater  the  smaller  the  distance  between  anode  and  filament. 
On  the  other  hand,  it  was  shown  above  that  a  decrease  in  this 
distance  decreases  the  voltage  drop  in  the  valve.  Hence  to  keep 
this  voltage  drop  small  the  valve  has  to  be  designed  so  as  to  pre- 
vent the  filament  and  anode  from  being  short-circuited  by  the 
strain. 

Arcing  across  the  glass  is  another  factor  which  has  to  be 
reckoned  with  in  designing  high  voltage  valves.  Fig.  46  shows  a 
General  Electric  valve  designed  by  Dushman  to  rectify  100,000 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE     121 

volts.1  Arcing  across  the  glass  is  prevented  by  placing  anode  and 
filament  terminals  at  opposite  ends  of  the  elongated  parts  of  the 
tube.  The  elongation  necessary  for  the  use  of  such  high  voltages 
necessitates  special  means  of  rigidly  supporting  the  electrodes. 

For  lower  voltages  the  distance  between  the  terminals  can  be 
smaller,  which  greatly  facilitates  the  construction.     A  simple  type 


FIG.  46. 

of  valve  is  shown  in  Fig.  47.  The  re-entrant  tubes  are  made  long 
enough  to  give  rigid  support  to  the  electrodes.  The  anode  can  be 
in  the  form  of  two  parallel  plates  placed  on  either  side  of  the 
filament  or  in  the  form  of  a  cylinder  or  preferably  a  flattened  cylin- 
der completely  surrounding  the  filament. 


FIG.  47. 

For  voltages  not  exceeding  a  few  hundred  volts  both  electrodes 
can  be  supported  from  the  same  press,  as  shown  in  Fig.  48.  This 
is  desirable  because  the  whole  structure  can  then  be  assembled 
and  the  spacing  accurately  adjusted  before  sealing  it  into  the 
bulb. 

If  the  voltage  exceeds  a  few  hundred  volts  sparking  may  take 
place  between  the  wires  in  the  press.  This  deleterious  effect  is 
i  General  Electric  Rev.,  p.  156,  March,  1915. 


122 


THERMIONIC  VACUUM  TUBE 


FIG.  48. 


mainly  due  to  the  anode  becoming  so  hot  that  it  volatilizes  and 

forms  a  metallic  deposit  on  the  press. 
It  can  to  some  extent  be  overcome  in 
the  tube  shown  in  Fig.  49.  In  this 
tube  both  electrodes  are  mounted  on 
the  same  re-entrant  tube  but  the  leak- 
age path  along  the  glass  between  the 
electrodes  is  considerably  lengthened  by 
taking  the  anode  lead  through  the  side 
tube  PA  which  is  closed  by  a  small  press 
at  P.  B  is  simply  a  glass  rod  to  give 
added  support  to  the  anode.  The  pos- 
sibility of  sparking  in  this  tube  depends 
almost  entirely  on  the  air  space  between 
the  wires  in  the  re-entrant  tube  and 
on  the  insulation  of  the  base  plate  CD. 

The  glass  press  can  also  be  protected  against  metallic  deposit  by 

means  of  a  metallic  shield  placed 

near  the  press,  as  was  for  example 

done  by  Dushman.1 

Although  the  voltage  drop  in 

the  valve   when    current   passes 

is  usually  not  high  and  should 

never  exceed  the  optimum  volt- 
age, which    generally    does    not 

amount    to    more    than    a    few 

hundred  volts,  the  full  voltage  of 

the  generator  is  impressed  on  the 

valve  during    the  blocking  half 

period.     The  tube  must  therefore 

be  evacuated  to  such  an  extent 

that  the  high  voltage  does  not 

start    a   glow    discharge.      This 

necessitates    clearing    the    elec- 
trodes and  walls  of  the  bulb  of 

gases  during    evacuation   to    an 

extent  depending  on  the  tempera- 
ture to  which    these  parts   are 

heated  during  operation  of  the  tube. 


FIG.  49. 


If  the  anode  is  of  tungsten 
1  U.  S.  Pat.  1,287,265. 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE     123 

the  power  that  could  safely  be  dissipated  in  it  is  about  10  watts  per 
cm2.1  This  corresponds  to  a  temperature  of  about  1600°  K.  If  the 
anode  temperature  rises  above  this  value  the  anode  emits  electrons 
at  a  sufficiently  high  rate  to  make  the  rectification  imperfect. 
If  the  anode  is  of  nickel  the  safe  power  dissipation  in  it  is  about  5 
watts  per  cm2.  The  size  of  the  anode  must  therefore  be  chosen  in 
proportion  to  the  power  loss  in  the  rectifier.  This  is  also  the  case 
with  the  size  of  the  glass  container.  Of  the  power  radiated  from 
the  internal  structure,  part  is  transmitted  through  the  glass, 
and  part  is  absorbed  and  radiated.  This  part  causes  a  heating 
of  the  glass.  The  size  of  the  glass  tube  or  bulb  must  be  so 
chosen  that  its  area  in  square  inches  is  not  less  than  the  total 
power  in  watts  dissipated  by  the  internal  structure.  This 
includes  power  radiated  from  the  filament  as  well  as  from  the 
anode. 

49.  Rectification  Efficiency.  Let  us  consider  the  operation  of 
the  thermionic  valve  in  a  circuit  such  as  that  shown  in  Fig.  50 
in  which  the  valve  is  supposed  to 
be  connected  to  a  source  of  con- 
stant a-c.  voltage  e.  During  the 
half  cycle  that  the  filament  is 
positive  with  respect  to  the  an- 
ode no  current  is  transmitted  by 
the  valve  and  the  potential  dif- 
ference established  between  its 
terminals  is  equal  to  the  full 
voltage  supplied  by  the  generator. 
During  the  other  half  cycle  elect- 
rons emitted  from  the  hot  -filament 

are  driven  to  the  anode  and  current  flows  in  the  circuit  FPGD. 
The  voltage  drop  in  the  valve  now  depends  upon  its  resistance  and 
the  load  resistance  r.  Suppose  that  r  is  initially  so  large  that  the 
voltage  drop  E  in  the  valve  is  less  than  the  optimum  voltage  E8. 
If  r  be  now  decreased  the  current  increases  and  the  voltage  drop  in 
the  rectifier  also  increases.  How  this  change  takes  place  can 
readily  be  seen  by  considering  the  d-c.  values.  Suppose  a  direct 
voltage  equal  to  E0  be  applied  instead  of  the  alternating  voltage, 
the  filament  being  negative  with  respect  to  the  anode.  If  /  be  the 

1  S.  DUSHMAN,  General  Electric  Review,  loc.  cit. 


124  THERMIONIC  VACUUM  TUBE 

current  and  E  the  voltage  drop  in  the  valve  we  have  E  =  Eo—Ir. 
Putting  I  =  CE*  we  get: 


CE 


This  shows  that  as  r  is  decreased  the  voltage  drop  E  in  the  valve 
increases  slowly  at  first  and  then  more  rapidly.  But  this  equa- 
tion only  holds  until  E  becomes  equal  to  the  optimum  voltage  drop 
in  the  valve  and  the  current  obtained  is  equal  to  the  saturation 
current.  If  the  resistance  r  be  still  further  decreased  there  is  no 
further  increase  in  current  and  the  voltage  drop  in  the  valve  in- 
creases rapidly  and  may  cause  a  blow  out  of  the  tube.  By  suit- 
ably choosing  the  external  resistance  r  the  valve  can  be  made  to 
rectify  extremely  high  voltages. 

It  is  seen  that  the  thermionic  valve  acts  not  only  as  a  rectifier 
but  also  as  a  current  limiting  device.  When  the  filament  is  positive 
the  emitted  electrons  are  returned  to  it  and  there  is  then  con- 
sequently no  current.  When  the  filament  is  negative  the  current 
is  limited  by  the  saturation  value  which  it  cannot  exceed  and  which 
is  determined  by  the  total  number  of  electrons  emitted  per  second 
from  the  filament  at  the  temperature  used. 

To  obtain  an  expression  for  the  rectification  efficiency  let  us 
consider  the  general  case  by  supposing  that  the  valve  does  not 

completely  block  current  during 
the  blocking  half  period.  The 
current  wave  will  then  have  a 
shape  somewhat  like  the  curve 
Ji/2  (Fig.  51).  When  the  valve 
is  short-circuited  the  current 
through  the  external  resistance  r 
is  given  by  the  curve  IT.  The 
introduction  of  the  valve  not  only 
limits  the  current  in  both  half 
FIG.  51.  periods  due  to  the  addition  of  its 

resistance  in  the  circuit,  but  it  also 

changes  the  shape  of  the  current  wave  due  to  its  non-linear 
current-voltage  characteristic.  Let  D  (Fig.  50)  be  an  a-c.  measur- 
ing instrument  such  as  a  dynamometer  and  G  a  galvanometer  to 
read  the  d-c.  component  of  the  current.  Let  IQ  be  the  reading 
of  the  a-c.  instrument  when  the  valve  is  in  the  circuit  and  i  the 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE     125 
reading  of  the  galvanometer.     The  efficiency  is  sometimes  defined 

<j 

simply  as  the  ratio  —  .     To  see  what  this  means,  let  I\  be  the 
^o 

maximum  instantaneous  value  of  the  current  in  the  transmission 
half  period  and  h  the  corresponding  maximum  in  the  blocking 
half  period.  Then  the  dynamometer  reading  i$  will  be  given  by 


and  the  galvanometer  reading  by 


where  g  is  the  "  amplitude  factor/'  given  by  the  ratio  of  the 
R.M.S.  to  the  maximum  values  of  the  current,  and  /  is  the  "  form 
factor,"  given  by  the  ratio  of  the  R.M.S.  to  the  true  mean  current.1 
Both  g  and  /  depend  upon  the  shape  of  the  wave.  For  a  pure  sine 
wave  </  =  .707  and  /=1.111.  The  above  expression  for  the  ef- 
ficiency therefore  takes  the  form 


= 

to    / 


From  this  it  is  seen  that  if  the  valve  conducts  absolutely  no  cur- 
rent during  one-half  period,  i.e.,  /2  =  0  and  if  the  current  wave 
shape  during  the  other  half  period  is  that  of  a  pure  sine  wave, 
i.e.,  /=  1.111,  the  efficiency  as  expressed  above  has  a  maximum 
value  of  90  per  cent. 

Instead  of  this  method  Fleming  2  makes  use  of  tne  expression 

-^ — -  and  calls  it  the  "  rectifying  power."     By  a  simple  trans- 
/i 

formation  of  equation  (17)  this  can  be  expressed  as: 

•*•  1       •*  2          ~7  /i  o\ 

— j — =-H-.   .......     (18) 


1  J.  A.  FLEMING,  "  Alternating  Current  Transformer,"  Vol.  I  D.  585. 

2  J.  A.  FLEMING,  Proc.  Roy.  Soc.,  Jan.,  1905,  p.  484. 


126  THERMIONIC  VACUUM  TUBE 

This  equation  expresses  the  ratio  of  unidirectional  current 
observed  to  the  unidirectional  current  that  would  flow  if  the 
rectification  produced  by  the  valve  were  complete.  It  is  seen 

that  if  ~  is  equal  to  the  form  factor/,  that  is  if  1 2  =  0  (equation  17), 

this  ratio  becomes  unity,  or  100  per  cent,  irrespective  of  the  shape 
of  the  wave. 

It  is,  however,  not  sufficient  to  know  how  well  the  device  blocks 
current  in  the  one  direction.  It  is  equally  important  to  know  how 
much  current  it  transmits  in  the  other  direction,  and  this  depends 
upon  the  wave  shape  of  the  transmitted  current.  Furthermore, 
the  insertion  of  the  valve  in  the  circuit  is  equivalent  to  the  intro- 
duction of  an  extra  resistance.  Let  J  be  the  dynamometer  read- 
ing when  the  valve  is  short-circuited,  i.e.,  the  R.M.S.  value  of 
the  current  AI'BI'C  (Fig.  51).  Then  the  ratio  of  useful  rectified 
current  to  the  available  alternating  current  is  obtained  by  dividing 
equation  (16)  by  J  =  gl',  This  gives: 


J       2/7' 

If  EI  be  the  maximum  value  of  the  applied  generator  voltage 

Tjl 

(see  Fig.  50)   and  r  the  load  resistance,  we  have  7'  =  —    and 

r 

77F 

7i  —          ,  where  r\  is  the  resistance  of  the  valve  at  the  maximum 

voltage  across  its  terminals.  The  resistance  of  the  valve  for  any 
given  voltage  between  filament  and  anode  is  given  by  the  ratio 
of  that  voltage  to  the  current  and  is  a  function  of  the  voltage. 
Referring  to  the  characteristic  curve  (Fig.  42)  it  will  be  evident 
that  as  the  voltage  increases  from  zero  to  the  optimum  value  Es, 
the  resistance  decreases  from  infinity  to  a  definite  value  given  by 
the  reciprocal  of  the  slope  of  the  straight  line  joining  A  and  0. 
What  we  are  concerned  with  here  is  the  resistance  which  obtains 
when  the  voltage  has  its  maximum  value.  We  shall  refer  to  this 
as  the  minimum  resistance  and  denote  it  by  n.  Combining  the 
equations  for  I'  and  7i  we  obtain 


r= 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE    127 


and  hence  with  the  help  of  equations  (18)  and  (19): 
*  1 


(20) 


If  the  tube  conducts  no  current  in  the  one  direction  the  form 
factor  /  is  equal  to  -?,  the  ratio  of  the  dynamometer  to  the  galvano- 

meter reading  when  the  valve  is  in  the  circuit.  This  follows  from 
equation(17)  for  /2  =  0.  If  furthermore  r\  is  negligibly  small 
compared  with  the  load  resistance  r,  equation  (20)  reduces  to 


= 

J~2f 

Hence  the  rectification  efficiency  as  given  by  equation  (20)  would 
reach  a  maximum  of  50  per  cent  (1)  if  the  form  factor  /  were 
unity;  (2)  if  the  minimum  resistance  of  the  valve  is  negligibly 
small  compared  with  the  load  resistance  and  (3)  if  the  valve 
completely  blocks  current  in  one  direction.  Let  us  see  how 
nearly  these  conditions  can  be  satisfied. 

(1)  As  regards  the  first  condition,  the  form  factor  is  always 
greater  than  unity  for  thermionic  valves.  If  the  relation  between 
the  applied  voltage  and  the  current  were  linear  and  a  true  sinusoi- 
dal voltage  be  impressed  on  the  valve  the  current  loop  during  the 
transmission  half  period  would  also  be  sinusoidal  and  then  the  form 
factor  would  be  that  of  a  sinusoid,  namely  1.111.  If  this  is  not 
the  case  but  if  the  current  is,  let  us  say,  proportional  to  the  nth 
power  of  the  applied  voltage  the  current  wave  will  have  a  shape 
somewhat  like  that  shown  in  Fig.  43  (p.  116),  and  the  form  factor 
/  must  be  determined  from  the  root  mean  square  and  true  mean 
values  of  sinn  pt.  For  such  cases  /  and  the  amplitude  factor  g 
can  be  obtained  from  a  table  of  gamma  functions  by  expressing 
them  in  the  forms: 


(22) 


128 


THERMIONIC  VACUUM  TUBE 


The  following  table  gives  the  values  of  /  and  g  for  a  range 
of  values  of  the  exponent  n: 


Exponent  n. 

Form  Factor,  /. 

Amplitude  Factor  g. 

1.00 

1.111 

.707 

1.25 

1.141 

.677 

1.50 

1.170 

.652 

1.75 

1.199 

.632 

2.00 

1.225 

.612 

2.50 

1.275 

.584 

3.00 

1.320 

.560 

To  use  the  above  expression  for  the  rectification  efficiency  it  is 
necessary  first  to  determine  the  relation  between  the  total  applied 
voltage  and  the  current  through  the  valve  and  external  resistance. 
To  a  first  approximation  this  can  be  expressed  by  a  simple  power 
relation  in  which  case  the  exponent  n  of  the  voltage  can  be  deter- 
mined by  plotting  the  logarithms  of  the  voltages  against  the  logar- 
ithms of  the  currents.  The  corresponding  values  of  /  and  g  can 
then  be  obtained  from  the  above  table. 

In  this  connection  it  must  be  pointed  out  that  there  is  a  differ- 
ence between  the  current  voltage  characteristic  of  the  valve  itself 
and  that  of  the  circuit  comprising  the  valve  and  external  resist- 
ance. The  effect  of  this  resistance  is  to  straighten  out  the  char- 
acteristic, because  it  means  the  addition  of  an  ohmic  resistance 
which  partially  neutralizes  the  curvature  of  the  characteristic  due 
to  the  non-ohmic  resistance  of  the  valve.  The  effect  is  obviously 
the  more  marked  the  larger  the  external  resistance  compared 
with  the  valve  resistance.  However,  since  the  applied  voltage 
alternates  the  valve  resistance  continually  changes.  During  the 
half  period  when  the  plate  is  negative  with  respect  to  the  filament 
the  resistance  of  the  valve  is  infinite,  and  during  the  other  half 
period  the  resistance  decreases  from  infinity  to  a  definite  minimum 
value  and  then  increases  again  to  infinity.  Hence,  for  low  in- 
stantaneous values  of  the  applied  alternating  voltage  the  external 
resistance  does  not  exert  a  marked  effect  in  straightening  out  the 
characteristic,  so  that  even  if  the  external  resistance  is  large  the 
current  still  tends  to  assume  a  shape  somewhat  like  that  shown 
in  Fig.  43  although  on  the  whole  it  approximates  more  closely 
to  the  shape  of  the  sinusoid. 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE     129 

(2)  Considering  now  the  effect  on  the  efficiency  as  given  by 
equation  (20)  of  the  relative  values  of  the  external  or  load  resist- 
ance to  the  minimum  resistance  n  of  the  valve  itself,  it  is  to  be 
noted  that  when  the  valve  is  used  to  rectify  high  voltages  its 
minimum  resistance  is  generally  negligibly  small  compared  with 
the  load  resistance.     It  was  shown  in  Section  47  that  the  valve  is 
operated  most  efficiently  when  the  peak  value  of  the  voltage  applied 
to  its  terminals  is  just  equal  to  the  minimum  voltage  necessary  to 
give  the  saturation  current.     This  we  called  the  optimum  voltage 
and  it  is  given  by  Es  in  Fig.  42.     If,  for  example,  the  saturation 
current  of  the  valve  at  a  certain  filament  temperature  is  300  milli- 
amperes  and  if  the  valve  is  so  designed  that  the  minimum  voltage 
across  its  terminals  necessary  to  give  this  current  is  150  volts,  the 
minimum  valve  resistance  is  500  ohms.      Now,  if  the  peak  value 
of  the  voltage  to  be  rectified  is  30,000  volts  (about  21,000  volts 
effective)  the  total  resistance  of  the  circuit  must  be  100,000  ohms, 
which  is  very  large  compared  with  that  of  the  valve.     For  higher 
voltages  the  device  operates  even  more  efficiently,  so   that  the 
second  factor  in  the  denominator  of  equation  (20)  reduces  to 
unity. 

(3)  Coming  to  the  third  condition  mentioned  we  can  for  most 
practical  purposes  regard  the  thermionic  valve  as  a  perfect  uni- 
lateral device;    it  completely  blocks  current  in  one  direction, 
provided  that  the  plate  does  not  become  so  hot  that  it  emits  an 
appreciable  number  of  electrons 

and  provided  that  the  valve  be 
so  constructed  that  leakage  be- 
tween its  terminals  across  the 
glass  does  not  take  place.  At 
high  frequencies,  however,  the 
rectification  becomes  imperfect 
due  to  the  capacity  between  the 
electrodes  (see  p.  134). 

The  completeness  of  the  recti- 
fication produced  by  thermionic  ~"FIG.  52. 
valves  is  shown  in  the  oscillogram 

given  in  Fig.  52  and  which  was  obtained  by  Dushman.1  The 
upper  curve  gives  the  voltage  across  the  tube  and  the  lower  curve 
the  rectified  current. 

1 S.  DUSHMAN,  General  Electric  Review,  loc.  cit. 


130  THERMIONIC  VACUUM  TUBE 

Assuming,  therefore,  that  the  load  resistance  is  so  large  that 
we  can  regard  the  characteristic  of  the  circuit  as  linear,  we  have 

f=  i.in  and  —  =  0.     If  the  valve  completely  blocks  current  in 
one  direction,  ^?  =/.     Putting  these  values  into  equation  (20)  we 

& 

find  that  the  highest  efficiency  obtainable  is  45  per  cent. 

If  the  resistance  of  a  valve  during  one  half  period  is  infinite, 
and  during  the  other  half  period  zero,  the  valve  is  perfect,  and  the 
rectification  efficiency,  as  expressed  by  equation  (20),  becomes 
independent  of  the  resistance  r,  used  in  the  external  circuit, 
because  under  these  conditions  n  is  zero.  Actually,  however, 
the  resistance  of  valves  during  the  transmission  half  period  is 
not  zero,  so  that  according  to  equation  (20)  the  efficiency  increases 
the  larger  the  external  resistance  r  becomes  in  comparison  with  the 
resistance  r\  of  the  valve.  Now,'  it  was  explained  in  Section  47 
that  for  most  efficient  operation  the  voltage  drop  in  the  valve 
during  the  transmission  half  period  should  never  exceed  the  value 
necessary  to  give  the  saturation  current.  This  means  that  the 
external  resistance  should  be  so  adjusted  that  the  maximum 
voltage  drop  across  the  valve  is  equal  to  the  optimum  voltage. 
In  other  words,  the  ratio  of  n  to  r,  occurring  in  the  second  ex- 
pression in  the  denominator  of  (20)  becomes  smaller  the  larger 
the  voltage  that  is  to  be  rectified.  The  rectification  efficiency 
of  the  valve,  therefore,  increases  with  the  voltage  that  is  to  be 
rectified  and  approaches  the  maximum  efficiency  that  could  be 
obtained  with  a  perfect  valve. 

The  efficiency  can,  of  course,  be  doubled  by  making  use  of  both 
half  waves  so  that  the  rectified  current  is  given  by  the  continuous 
lines  of  Fig.  53  instead  of  Fig.  43.  This  can  be  done  by  using 
two  tubes  in  the  circuit  shown  in  Fig.  54.  It  is  seen  that  for  both 
half  periods  the  electron  current  in  the  load  is  in  the  direction 
of  the  arrow.  This  scheme  necessitates  dividing  the  input  voltage 
on  the  secondary  side  of  the  transformer  T.  In  order  to  make 
use  of  the  full  voltage  of  the  transformer  the  circuit  shown  in  Fig. 
55  can  be  used.1  Here  again  current  flows  in  the  load  resistance 
during  both  half  periods  in  the  direction  of  the  arrow.  Such  an 
arrangement  requires  four  times  the  total  power  necessary  to  heat 
the  filaments.  The  filament  heating  power  is,  however,  small 

1  GRAETZ,  Die  Elektrizitat  uncl  ihre  Anwendungen,  15th  Edition,  p.  444. 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE    131 


compared  with  the  power  that  becomes  available  in  the  form  of 
unidirectional  current.  This  is  the  more  marked  the  higher 
the  voltage  that  is  to  be  rectified.  Let  us,  for  example,  take  the 
case  considered  above  in  which  a  tube  that  could  give  300  milli- 
amperes  was  made  to  rectify  21,000  volts.  Referring  to  the 


^ 


T 


\ 


\j 

FIG.  53. 


Load 


FIG.  54. 


table  on  p.  77  it  is  seen  that  if  the  filament  is  of  tungsten  and  is 
operated  at  a  temperature  a  little  under  2500°  K.  the  thermionic 
efficiency  is  10  m.a.  per  watt.  Since  the  necessary  saturation  cur- 
rent 1 1  is  300  m.a.  the  power  necessary  for  heating  the  filament  of 
the  valve  under  consideration  is  30  watts.  The  power  available 
in  the  form  of  unidirectional  current  in  the  load  can  be  obtained 


FIG.  55. 


from  equation  (15).     Putting  /2  =  0,  the  heating  equivalent  of  the 
current  in  the  load  resistance  is  g  -£•  and  the  available  power  in 


the  form  of  unidirectional  current  is  g2 


Ifr 


Assuming  that  the 


current-voltage    characteristic  of  the  circuit  is,  in  view  of  the 
high  load  resistance  of  about  100,000  ohms,  practically  linear, 


132  THERMIONIC  VACUUM  TUBE 

we  can  put  #  =  .707.  This  makes  the  available  power  about 
IKW,  which  is  quite  large  compared  with  the  power  necessary  to 
heat  the  filament.  It  is,  in  fact,  so  much  larger  that  it  is  advan- 
tageous to  quadruple  the  filament  heating  power  in  order  to  double 
the  output  power  with  the  arrangement  shown  in  Fig.  55. 

Now,  what  is  ordinarily  observed  in  practice  is  not  the  heating 
current  IQ  which  must  be  measured  with  an  a-c.  meter,  but  the 
true  mean  of  the  unidirectional  current  i  measured  with  an 
ordinary  d-c.  ammeter.  The  output  power  can  then  be  readily 
obtained  from  i2/2r,  where  /  is  the  form  factor  (equation  17), 
assuming  that  the  valve  does  not  conduct  current  at  all  in  one 
direction,  an  assumption  which  is  justified  in  most  practical  cases. 

50.  Production  of  Constant  Source  of  High  Voltage  with  the 
Thermionic  Valve.  As  a  rectifier  the  thermionic  valve  offers 
three  distinct  advantages:  It  can  be  used  to  rectify  voltages  rang- 
ing up  to  the  highest  met  with  in  practice;  it  rectifies  currents 
of  comparatively  high  frequency  as  well  as  low  frequency  currents; 
it  completely  blocks  current  in  one  direction,  provided  the  fre- 
quency is  kept  below  a  certain  limit  depending  on  the  voltage  that 
is  to  be  rectified.  (This  effect  will  be  discussed  below.)  The  value 
of  these  three  advantages  will  become  apparent  in  the  following 
discussion : 

When  used  as  a  rectifier  under  the  conditions  described  in 
the  previous  paragraphs  the  valve  produces  a  unidirectional  pul- 
sating current.  In  some  practical  applications,  such  as  direct 
current  high  voltage  transmission,  festing  of  dielectric  strength 
of  insulators  at  high  voltages,  use  of  a  d-c.  source  of  high  voltage 
for  laboratory  purposes,  etc.,  it  is  necessary  that  the  pulsating 
current  be  smoothed  out  into  a  constant  direct  current.  We  shall 
therefore,  proceed  to  a  discussion  of  the  means  whereby  this 
smoothing  out  can  be  accomplished. 

When  the  required  direct  current  is  small  and  it  is  not  essential 
that  the  wave  be  completely  smoothed  out,  we  can  resort  to  the 
simple  and  well-known  expedient  of  shunting  the  load  with  a 
sufficiently  large  condenser,  C\  (Fig.  56).  This  condenser  acts  as  a 
reservoir  from  which  a  practically  constant  current  can  be  drawn 
continuously,  it  being  charged  up  in  alternate  half  periods  and 
always  in  the  same  direction.  The  effect  of  this  condenser  (the 
inductance  L  being  omitted)  can  be  seen  from  Fig.  57.1 

1  A.  W.  HULL,  General  Electric  Rev.,  Vol.  19,  p.  177,  1916. 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE     133 

The  sine  wave  represents  the  transformer  voltage  and  the 
heavy  line  the  output  voltage  across  the  condenser,  which  initially 
becomes  charged  up  to  the  full  peak  value  of  the  input  voltage. 
During  the  rest  of  the  period  it  receives  no  charge  until  the  input 
voltage  becomes  greater  than  the  value  to  which  the  condenser 
voltage  has  dropped  in  virtue  of  the  current  drain  from  it.  It  is 


FIG.  56. 

seen  that  the  advantage  offered  by  this  type  of  valve  that  it  almost 
completely  blocks  current  in  one  direction  is  an  important  one; 
the  condenser  never  discharges  itself  through  the  input  circuit 
CiTV.  The  rate  at  which  the  condenser  discharges  through 
the  load  is  given  by 


where  EI  is  the  voltage  across  the  condenser  plates  and  i  the 
average  current,  which  can  be  regarded  as  practically  constant. 


FIG.  57. 


Hence,  integrating  between  EA  and  EB  and  putting  EA—EB= 
the  voltage  variation  across  the  condenser  is: 


(23) 


134  THERMIONIC  VACUUM  TUBE 

where  t  is  almost  a  complete  period  and  can  in  the  following 
calculations  be  regarded  approximately  as  such.  We  can  therefore 
write  (23)  in  the  form 


ET 


(24) 


where  r  is  the  load  resistance  and  Er  the  direct  voltage  in  it. 
This  equation  shows  that  the  condenser  Ci  alone  will  appreciably 
reduce  the  voltage  fluctuation  provided  the  load  resistance  is  suf- 
ficiently large.  The  same  result  can,  of  course,  also  be  secured 
by  increasing  the  capacity  and  the  frequency  of  the  impressed 
generator  voltage.  If,  however,  the  load  resistance  is  small  C\ 
would  have  to  be  made  so  large  as  to  make  its  use  impracticable 
nor  can  the  frequency  be  made  very  high,  because  then  the  capac- 
ity of  the  tube  itself  would  become  effective  with  the  result  that 
the  tube  would  not  rectify  completely.  That  the  limiting  fre- 
quency is  lower  than  is  sometimes  assumed  will  be  seen  from  the 
following  simple  consideration. 

To  operate  the  valve  most  efficiently,  the  voltage  across  it 
during  the  transmission  half  period  must  not  exceed  the  optimum 
value,  which  is  generally  of  the  order  of  a  few  hundred  volts,  so 
that  in  rectifying  very  high  voltages,  say  100,000  volts,  the  load 
resistance  must  be  very  high.  But  it  must  always  be  small  com- 
pared with  the  resistance  of  the  valve  during  the  blocking  half 
period.  For  low  frequencies  or  d-c.  this  resistance  of  the  valve  is 
infinite,  but  at  high  frequencies  the  valve  may  on  account  of  its 
electrostatic  capacity  have  an  impedance  which  is  comparable  with 
the  load  resistance  and  then  currents  of  comparable  magnitude 
will  obviously  flow  in  both  directions  in  the  load  resistance. 
Thus,  if  the  capacity  of  valve  be  C  =  10  micro-microfarads,  which 
is  well  within  the  range  of  the  capacities  of  the  valves  used  in 
practice,  its  impedance  at  a  frequency  of  16,000  cycles  per  second 
is  1  megohm.  If  now  the  voltage  to  be  rectified  is  EQ  (peak  value), 
the  optimum  voltage  of  the  valve  Es  and  the  maximum  current 
obtainable  from  it  I5  then  the  load  resistance  r  is 

_Ep  —  Es 

I. 

E,  can  usually  be  neglected  in  comparison  with  E0.  If  the 
voltage  to  be  rectified  is,  say,  100,000  volts,  E0=  140, 000  volts, 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE     135 

and  if  7S=100  milliamperes,  r  will  be  1.5  megohms  (approx.). 
Hence  the  impedance  of  the  tube  due  to  its  electrostatic  capacity 
at  16,000  cycles  is  of  the  same  order  of  magnitude  as  the  load 
resistance  which  is  necessary  when  the  voltage  to  be  rectified  is 
of  the  order  of  100,000.  The  condenser  will  therefore  partly 
discharge  itself  in  alternate  half  periods  through  the  input  circuit 
CiTV  (Fig.  56).  At  lower  voltages  this  effect  is  not  so  marked, 
so  that  higher  frequencies  could  be  used  to  advantage. 

In  order  to  smooth  out  the  voltage  fluctuations  somewhat  more 
effectively  an  inductance  L  is  sometimes  inserted  in  series  with  the 
load,  as  shown  in  Fig.  56.  When  the  load  resistance  is  low  the 
inductance  helps  appreciably,  but  for  high  load  resistances  it  serves 
little  purpose.  Considering  this  inductance,  it  is  evident  that  the 
voltage  fluctuation  dEr  across  r  is  to  the  voltage  fluctuation  8Ei 
across  the  condenser  C\  as  the  ratio  of  r  to  the  impedance  of  L 
and  r  in  series.  Thus: 


It  is  seen  that  for  large  values  of  the  load  resistance  r  the  inductance 
contributes  little  to  smoothing  out  the  fluctuations,  but  it  helps 
appreciably  at  the  lower  load  resistances.  Assuming  that  the 
coil  L  is  a  pure  reactance,  the  d-c.  voltage  in  r  is  the  same  as  that 
across  the  condenser  C\.  Hence  the  percentage  voltage  fluctua- 
tion can  be  obtained  by  combining  equations  (24)  and  (25).  This 
gives 

(26) 


The  relation  between  -=^  and  log  r  is  shown  by  curve  //  of 

&r 

Fig.  58.  The  curve  /  gives  the  relation  when  the  inductance  L 
is  omitted.  These  curves  were  computed  with  the  following 
values:  L  =  100  henrys,  C=10~~9  farad.  The  curves  show  that 
if  the  load  resistance  is  greater  than  about  a  megohm,  that  is  for 
resistances  of  the  order  of  magnitude  used  when  rectifying  very 
high  voltages,  practically  the  same  result  can  be  secured  by  using 
only  the  condenser  instead  of  adding  the  inductance.  On  the 
other  hand  the  condenser  alone  is  useless  at  load  resistances  less 
than  a  megohm. 


135 


THERMIONIC  VACUUM  TUBE 


Better  results  can,  of  course,  be  obtained  by  adding  more 
sections  to  the  wave  filter  LCi.  Such  an  arrangement  is  shown  in 
Fig.  59,  which  at  the  same  time  shows  a  circuit  that  makes  possible 


IB 


I 

V 

0)4 


N 


3  4  5  6 

109  r  (Load  Resistance) 

FIG.  58. 


the  use  of  both  half  periods  by  employing  two  tubes.  It  will  be 
seen  that  current  flows  in  the  direction  of  the  arrow  (say)  when 
A  is  positive  and  B  negative  as  well  as  when  A  is  negative  and  B 


FIG.  59. 


positive,  the  current  being  transmitted  through  the  tubes  alter- 
nately. 

Let  us  now  see  to  what  extent  the  added  filter  section  con- 
tributes in  reducing  the  voltage  fluctuations  in  r  and  how  they 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE     137 


depend  on  the  value  of  r.     Let  8Er,  bEz  and  8Ei  be  the  voltage 
fluctuations  at  the  terminals  of  r,  €2  and  Ci  respectively.     Then 


(27) 


If  Z2  be  the  impedance  of  the  circuit  C^Ly,  as  measured  between 
the  terminals  of  C2,  and  Z\  the  impedance  of  LI  and  Z2  in  series, 
then 


Now  since 


1  —  C2L2  co2  -\-jrCz  co 

.     .     (28) 


we  get 

2' 


The  voltage  fluctuation  across  the  terminals  of  condenser  C\  is 
given  by  equation  (24).  Hence,  multiplying  together  (24), 
(27)  and  (29)  and  expressing  the  impedances  numerically  instead 
of  symbolically  the  ratio  of  the  voltage  fluctuation  in  r  to  the  d-c. 
voltage  in  r  is : 

M^  = — ^ g-^.     (30) 

A.  W.  Hull l  described  a  high  voltage  rectifying  set  in  which 
he  used  two  condensers  but  only  one  inductance,  LI.  Putting 
L2  =  0  in  equation  (30)  the  voltage  fluctuation  for  Hull's  set 
becomes : 

8Er  27T 


ET     coCi[r2(l-LiC2co2)2+Li2co2] 


1A 


zero).     .     (31) 


and  if  C2  =  0  this  equation  reduces  to  (26)  which  is  the  equation 
for  the  circuit  shown  in  Fig.  56.  On  the  other  hand,  if  the  induc- 
tance, frequency  and  second  condenser  have  such  values  as  to 
put  LI  and  C2  in  resonance,  i.e.,  when  LiC2co2=l,  then  the 

1  A.  W.  HULL,  loc.  cit.     Hull's  equations  are  not  the  same  as  these,    since 
he  did  not  add  the  imaginary  terms  in  quadrature. 


138  THERMIONIC  VACUUM  TUBE 

arrangement  corresponding  to  equation  (31)  is  worse  than  that  of 
Fig.  56  in  which  the  second  condenser  is  omitted  altogether.  It 
follows  that  for  this  circuit  to  be  better  than  that  of  Fig.  56  we  must 
make  LiC2<o2>2.  This  condition  is  easily  satisfied  in  practice. 
In  Hull's  set,  for  example,  the  values  of  LI,  €2  and  w  happen  to 
be  such  that  their  product  is  about  60.  However,  although  this 
circuit,  containing  one  inductance  and  two  capacities  is  a  decided 
improvement  over  the  simpler  one  shown  in  Fig.  56,  it  is  better  to 
split  the  inductance  and  use  the  circuit  of  Fig.  59.  This  circuit 
has  a  decided  advantage  at  lower  load  resistances,  even  when  the 
inductances  LI  and  L^  are  each  one-half  of  the  value  of  LI  when 

r-Tjl 

L/2  =  0.     The  percentage  ratio  of  — r  as  a  function  of  log  r  for  these 

Ur 

two  cases  is  shown  by  curves  III  and  IV  of  Fig.  58.  Curve  III 
was  computed  for  the  following  values:  Ci  =  C2  =  10-9  farad, 
LI  =  100  henrys,  co  =  27rX4000.  In  curve  IV  the  values  were  the 
same  except  that  Li  =  L2  =  50  henrys.  It  can  readily  be  seen 
that  a  frequency  of  4000  is  obtained  in  the  filter  circuit  when 
the  frequency  of  the  voltage  impressed  at  T  is  2000  cycles,  since 
by  using  two  tubes  as  shown  in  Fig.  59,  the  condensers  are  charged 
up  every  half  period  of  the  voltage  in  T. 

Curve  IV  shows  the  value  of  a  circuit  like  that  shown  in  Fig.  59, 
when  it  is  desired  to  have  a  rectifying  set  which  is  to  operate 
with  large  variations  in  the  load  resistance. 

It  will  be  evident  that  these  circuits  simply  represent  a  type  of 
wave  filter  which  is  supposed  to  filter  out  all  frequencies  except 
zero,  that  is,  the  direct  current.  The  waves  obtained  in  the  output 
of  these  circuits  comprise  not  only  the  fundamental  frequency 
that  we  considered  in  the  above  computations,  but  also  a  number 
of  harmonics  which  are  generally  weak  compared  with  the  funda- 
mental. It  will  be  evident  that  harmonics  must  necessarily  be 
present,  considering  that  the  wave,  which  has  the  form  shown  in 
Fig.  53  is  not  a  pure  sinusoid.  Such  a  wave  can  always  be  ex- 
pressed in  a  Fourier  series  (equation  1).  It  will  also  be  seen  from 

?Tjl 

the  nature  of  the  above  equations  for  -^-  that   the  harmonics 

LP 

will  be  damped  out  more  effectively  than  the  fundamental.     They 

were  therefore  left  out  of  consideration  in  the  above  calculations. 

Another  type  of  circuit  that  could  be  used  for  smoothing  out 

the  voltage  fluctuations  was  suggested  to  me  by  Mr.  T.  C.  Fry, 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE     139 

and  is  a  special  case  of  Campbell  filter  (Fig.  60).  It  has  the  ad- 
vantage that  the  capacities  and  inductances  necessary  are  rela- 
tively small,  which  is  always  a  good  thing  when  rectifying  very 
high  voltages  in  view  of  the  difficulty  of  constructing  condensers 
of  high  capacity  for  high  voltage  work. 


FIG.  60. 

The  characteristic  of  this  filter  is  seen  from  Fig.  61,  where  the 
current  attenuation  produced  by  the  filter  is  plotted  against  the 
frequency.  The  capacities  and  inductances  can  be  so  chosen 

that  the  fundamental  frequency,  — ,  is  that  which  gives  infinite 

ZTT 

attenuation.    This  frequency  will  therefore  not  be  present  in  the 


Frequency 

FIG.  61. 


load   resistance.     The   filter   would   transmit   lower   frequencies 

than  — ,  but  such  frequencies,  except  zero,  are  not  present  when 

2vr 

—  represents  the  fundamental.     Hence,  for  all  frequencies  below 


140  THERMIONIC  VACUUM  TUBE 

—  only  direct  current  is  transmitted.     The  higher  frequencies 

2-7T 

will  be  transmitted  and  they  will  be  present  in  the  form  of  harmon- 
ics. These  are,  however,  so  weak  that  when  attenuated  to  the 

extent  shown  by  the  curve  to  the  right  of  — ,  their  effect  in  the  load 

ZTT 

is  practically  nil.  It  will  be  evident  from  the  nature  of  the  attenu- 
ation curve  that  when  using  such  a  filter  the  frequency  of  the 
input  must  be  adjusted  rather  accurately  to  the  value  determined 
by  the  constants  of  the  filter. 

Figs.  59  and  60  show  only  two  filter  sections.  If  desired,  better 
results  can  be  obtained  by  adding  more  sections. 

Before  leaving  this  subject  let  us  discuss  briefly  the  relative 
value  of  a  few  types  of  circuits,  considering  mainly  the  arrange- 
ments of  the  valves  irrespective  of  the  type  of  filter  used  in  the 
output  circuit. 

The  circuit  shown  in  Fig.  59  is  arranged  to  make  use  of  both 
half  waves.  The  voltage  fluctuation  at  the  condenser  C\  will 
therefore  be  of  double  the  frequency  of  the  wave  supplied  through 
the  transformer  T.  The  potential  of  points  A  and  B  (Fig.  59)  will 
always  be  180°  out  of  phase,  but  only  when  they  are  positive  with 
respect  to  0  will  the  voltage  be  effective  in  charging  up  the  con- 
densers. If  the  potentials  of  A  and  B  be  represented  by  the  broken 
lines  A'  and  B'  (Fig.  62)  the  potential  fluctuation  at  the  condenser 


FIG.  62. 

will  be  represented  by  the  curve  CD,  which  possesses  a  fundamental 
whose  frequency  is  twice  that  of  the  waves  A'  and  B'.  This  is 
an  advantage  because  it  follows  from  equation  (32)  that  the  higher 
the  frequency,  the  more  effectively  will  the  fluctuation  be  smoothed 
out  by  the  filter.  On  the  other  hand  this  circuit  has  the  disad- 
vantage that  the  voltage  impressed  on  the  valves  is  only  half  that 
supplied  by  the  transformer.  In  order  to  use  the  full  transformer 
voltage  we  could  resort  to  the  arrangement  shown  in  Fig.  55, 
replacing  r  by  the  filter  and  load  resistance  shown  in  Fig.  59. 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE     141 

Fig.  63  shows  a  circuit  whereby  the  transformer  voltage  1  can 
be  doubled.  When  the  transformer  voltage  is  such  that  D  is  at  a 
positive  potential  with  respect  to  0,  an  electron  current  will  flow 
in  the  direction  of  the  arrow  through  the  valve  AD,  thus  charging 
the  condenser  C  such  that  A  is  positive  with  respect  to  0.  But 
during  this  half  period  no  current  will  flow  through  DB.  During 


D 


rL 


FIG.  63. 

the  next  half  cycle  current  flows  only  through  DB,  charging  B 
negatively  with  respect  to  0.  The  potential  difference  between 
A  and  B  (if  the  condensers  did  not  discharge  themselves)  would 
therefore  be  twice  the  transformer  voltage.  What  actually 
happens  is  that  the  one  condenser  discharges  through  the  load  while 
the  other  is  being  charged.  Hence  if  the  broken  line  (Fig.  64) 


FIG.  64. 

represents  the  potential  of  the  point  D  with  respect  to  0,  the  curves 
A  'A'  and  B'B1  will  represent  the  potentials  of  A  and  B  respect- 
ively with  regard  to  0.  The  potential  difference  between  A  and  B  is 
therefore  obtained  by  adding  the  curves  A'  and  Bf  and  is  given  by 
EE.  Thus,  although  the  condensers  are  charged  only  in  alternate 
1  H.  GREINACHER,  Verb.  d.  D.  Phys.  Gesell.,  Vol.  16,  p.  320,  1914. 


142  THERMIONIC  VACUUM  TUBE 

half  periods,  the  voltage  fluctuation  in  the  circuit  leading  to  the 
filter  is  double  the  frequency  of  the  impressed  voltage,  while  the 
mean  voltage  on  the  filter  is  approximately  twice  the  impressed 
voltage. 

It  will  be  observed  from  the  above  discussion  that  there  are 
various  ways  in  which  thermionic  valves  can  be  used  for  increas- 
ing frequency. 

For  laboratory  work  it  is  often  necessary  to  have  a  source  of 
fairly  high  constant  voltage  supplying  very  small  currents,  such 
as  would  be  needed,  for  example,  in  the  study  of  photo-electric 
phenomena,  ionization  of  gases  by  radium  or  X-rays,  measurement 
of  the  intensity  of  X-rays  with  the  ionization  chamber,  etc.  For 
such  purposes  the  thermionic  valve  could  be  used  to  replace  the 
rather  troublesome  high  voltage  batteries  frequently  used  in 
university  laboratories,  which  consist  of  a  large  number  of  min- 
iature storage  or  dry  cells.  In  fact,  the  high  voltage  desired 
could  be  obtained  from  any  standard  storage  or  dry  cell  battery 
of  a  few  volts,  which  forms  part  of  the  equipment  of  any  physi- 
cal laboratory,  by  connecting  the  primary  of  the  transformer  to 
the  low  voltage  battery  through  an  interrupter.  This  could,  for 
example,  be  done  by  using  a  small  Ruhmkorff  coil  with  an  ordi- 
nary hammer  break.  Since  the  desired  current  is  small  the  valves 
could  be  designed  to  operate  with  very  small  power  expenditure  in 
the  filament.  It  must,  however,  be  remembered  that  when  using 
the  device  for  the  purposes  mentioned,  the  load  resistance  is 
usually  very  high,  and  hence,  in  order  to  prevent  the  condensers 
from  discharging  through  the  valves  during  the  blocking  half  peri- 
ods, the  valves  should  be  designed  to  have  the  lowest  possible 
electrostatic  capacity  and  the  frequency  of  interruption  of  the 
primary  current  should  not  be  very  high. 

51.  The  Thermionic  Valve  as  a  Voltage  Regulator.  The 
rapid  increase  in  the  saturation  thermionic  current  with  increase 
in  the  filament  temperature,  or  filament  heating  current,  as  is 
shown  by  Richardson's  equation,  can  be  utilized  to  control  the 
voltage  of  the  generator  of  varying  speed.  A  scheme  whereby 
this  can  be  done,  and  which  was  devised  by  H.  M.  Stoller,  is  shown 
in  Fig.  65.  Here  the  tube  is  used  to  regulate  the  voltage  supplied 
by  a  wind-driven  generator  such  as  has  been  used  on  airplanes. 
The  generator  is  designed  to  supply  a  high  voltage  for  the  plate 
circuit  of  thermionic  tubes  and  a  low  voltage  for  heating  the 


RECTIFICATION  OF  CURRENTS  BY  THERMIONIC  VALVE     143 

filaments.     D  and  M  are  the  differential  and  main  field  windings 
of  the  generator.     The  thermionic  valve  is  inserted  as  indicated 


High 
Voltage 


(§)  Common 


Voltage 


FIG.  65 


FIG.  66. 


at  V.     The  characteristics  of  such  a  valve  are  shown  in  Fig.  66. 
Suppose,  now,  that  the  speed  of  the  generator  is  so  low  that  the 


144 


THERMIONIC  VACUUM  TUBE 


current  flowing  through  the  filament  of  the  valve  is  1.2  amperes. 
With  this  filament  current  the  thermionic  current  through  the  valve 
and  the  differential  winding  is  small  and  practically  the  full 
voltage  is  obtained.  If,  now,  the  speed  of  the  generator  increases, 
the  filament  current  of  the  valve  increases,  but  this  causes  a  pro- 
portionately much  greater  increase  in  the  thermionic  current 
which  flows  through  D.  Thus  Fig.  66  shows  that  a  small  change 
in  the  filament  current  of  from  1.2  to  1.4  amperes  causes  a  five- 


100 


4,000  6,000 

Revolutions  Per  Minute 

FIG.  67 


10,000 


12,000 


fold  increase  in  the  thermionic  current.  This  causes  a  decrease 
in  the  field  flux  of  the  generator,  thus  restricting  the  increase  in  the 
output  voltage. 

The  regulation  obtained  with  such  a  device  is  shown  in  Fig.  67. 
HH  and  LL  represent  the  high  and  low  output  voltages  as  a  func- 
tion of  the  speed  of  the  generator,  and  it  will  be  seen  that  although 
the  speed  changes  from  about  4000  to  over  12,000  R.P.M.,  the 
voltage  output  remains  practically  constant. 


-,:-- 


CHAPTER  VII 
THE  THERMIONIC  AMPLIFIER 

EXCEPT  for  the  derivation  in  Chapter  III,  of  a  few  fundamental 
relationships  that  govern  the  discharge  in  three-electrode  devices, 
we  have  so  far  considered  only  the  simple  type  of  device  containing 
two  electrodes.  The  physical  principles  underlying  the  thermionic 
tubes  discussed  in  the  previous  chapters  are  applicable  also  to  the 
three-electrode  type  of  thermionic  tube  which  it  is  our  purpose  to 
treat  in  this  and  the  following  chapters.  This  device  consists 
essentially  of  a  highly  evacuated  vessel,  containing  a  thermionic 
cathode,  usually  in  the  form  of  a  filament  which  can  be  heated  by 
passing  a  current  through  it,  an  anode  and  a  discharge-controlling 
electrode  which  generally  takes  the  form  of  a  wire  mesh  or  grid, 
and  placed  between  the  cathode  and  anode.  This  third  electrode 
can,  however,  be  of  any  form,  since  a  controlling  effect  on  the 
discharge  can  be  obtained  by  so  positioning  a  conductor  with 
respect  to  the  path  of  the  discharge  that  potential  variations 
applied  to  it  will  cause  variations  in  the  current  flowing  between 
cathode  and  anode.  The  controlling  electrode  may,  for  example, 
be  in  the  form  of  a  plate  placed  on  the  side  of  the  cathode  opposite 
to  that  of  the  anode  or  in  the  form  of  a  wire  or  a  plurality  of  wires 
galvanically  connected  and  placed  in  the  plane  of  the  cathode 
parallel  to  that  of  the  anode.  The  theory  of  operation  of  the 
device  to  be  given  in  the  following  applies  to  these  various  struc- 
tures, but  will  be  explained  with  particular  reference  to  the  case  in 
which  the  auxiliary  or  discharge-controlling  electrode  takes  the 
form  most  commonly  used  in  practice,  namely,  a  grid  placed 
between  cathode  and  anode.  This  was  suggested  by  Lee  de 
Forest.1  Originally  he  used  the  device,  which  he  called  the 
"  audion  "  as  a  radio  detector.  It  has  since  developed,  however, 
that  its  use  is  not  by  any  means  limited  to  this  field,  it  being  now 
1 U.  S.  Patents  No.  841387,  1907;  No.  879532,  1908. 


146 


THERMIONIC  VACUUM  TUBE 


used  extensively  also  as  amplifier,  oscillation  generator,  and  in 
a  large  number  of  widely  varying  applications.  Fig.  68  shows  a 
commercial  type  of  thermionic  amplifier. 

52.  Action  of  the  Auxiliary  Electrode.  It  was  shown  in 
Chapter  III,  page  42,  that  the  relation  between  the  electron 
current  to  the  anode  or  plate,  and  the  potentials  applied  to  the 
grid  and  plate  with  respect  to  the  filament,  can  be  expressed  in  a 
simple  way  by  making  use  of  the  writer's  linear  stray  field  relation : 


(1) 


which  means  that  if  the  grid  and  filament  be  at  the  same  potential, 
a  potential  difference  Ep  between  filament  (or  grid)  and  plate, 

_^ <        causes  a  stray  field  to  act  through 

the  openings  of  the  grid  which  is 
equivalent  to  the  field  that  would  be 
produced  if  a  potential  difference 

equal  to  —  were   applied   directly 

between  the  filament  and  a  plane 
coincident  with  that  of  the  grid. 
The  small  quantity  e  represents  an 
intrinsic  potential  difference  be- 
tween the  filament  and  the  system 
constituted  by  the  grid  and  plate. 
The  constant  /x  depends  on  the 
structure  of  the  device  (see  p.  226). 
If  we  now  apply  a  potential  differ- 
ence Eg  directly  between  filament 
and  grid,  the  effective  voltage 
in  the  tube  is  obtained  simply 
by  adding  Es  and  Ett,  and  the 
current  can  be  expressed  as  a  function  of  this  sum,  thus 


FIG.  68. 


Before  discussing  this  relationship,  let  us  look  more  fully  into 
the  functions  of  the  two  quantities  Es  and  E0.  To  simplify  matters 
somewhat  we  shall  neglect  the  effect  of  the  small  quantity  e. 
In  Fig.  69,  the  distribution  of  the  field  intensity  in  the  region 


THE  THERMIONIC  AMPLIFIER 


147 


between  cathode  and  anode  of  a  three-electrode  device  is  repre- 
sented by  means  of  lines  of  force.  The  anode  is  assumed  to 
remain  at  a  constant  positive  potential  with  respect  to  the  cathode, 
the  potential  of  which  we  can  call  zero.  The  three  diagrams  shown 
refer  to  the  cases  in  which  the  potential  of  the  grid  is  positive,  zero 
and  negative.  Looking  upon  the  intensity  of  the  field  as  the  num- 
ber of  lines  of  force  passing  through  unit  area,  it  will  readily  be 
seen  in  a  general  way  how  the  potential  of  the  grid  affects  the 
flow  of  electrons  from  the  cathode. 

But  before  considering  the  flow  of  electrons  it  must  be  pointed 
out  that  the  diagrams  in  Fig.  69  represent  the  distribution  of 
field  intensity  only  for  the  case  in  which  the  space  between  cathode 


6 

FIG.  69. 


and  anode  is  free  from  any  dislodged  electric  charges.  As  soon 
as  charges  are  introduced,  such  as  electrons  moving  from  cathode 
to  anode,  some  of  the  lines  of  force  proceeding  from  the  anode  will 
end  on  the  electrons,  and  hence  the  density  of  the  lines  of  force, 
i.e.,  the  intensity  of  the  field  or  the  potential  gradient,  will  be 
greater  near  the  anode  and  less  near  the  cathode  than  indicated 
in  Fig.  69.  This  space  charge  effect  can  be  made,  clearer  by 
representing  the  field  intensity  as  shown  in  Figs.  70  and  71.  Fig. 
70  shows  the  case  in  which  there  are  no  electrons  in  the  space, 
such  as  would  be  the  case  if  the  cathode  were  cold.  BP  repre- 
sents the  potential  of  the  anode,  that  of  the  cathode  being  zero. 
The  potential  gradient  (or  field  intensity)'  is  given  by  the  slope 
of  the  lines  PaO,  etc.  It  is  easily  seen  that  the  field  between 


148 


THERMIONIC  VACUUM  TUBE 


cathode  and  grid  is  the  resultant  of  Es  and  Eg.  The  lines  PaO, 
PbO  and  PcO  therefore  represent  the  distribution  of  field  intensity 
for  the  three  cases  in  which  Es+Eg  is  greater  than,  equal  to  or  less 
than  zero. 

If,  now,  the  cathode  be  hot  enough  to  cause  a  copious  emission 
of  electrons  from  it,  the  field  intensity  is  no  longer  a  linear  function 
of  the  distance  between  cathode  and  anode,  but  can  be  represented 
in  a  rough  way  by  the  curves  shown  in  Fig.  71.  If  E,+Eg>0, 
the  field  distribution  can  be  represented  somewhat  by  the  curve 
OaP.  If  E5+EofO,  the  field  between  the  cathode  and  the 
equivalent  grid  plane  is  negative  and  the  emitted  electrons  are 


FIG.  70. 


FIG.  71. 


returned  to  the  cathode.  The  curvature  of  the  lines  06  and  Oc 
is  due  to  the  initial  velocities  of  the  electrons  (see  Fig.  21). 

The  lines  of  force  proceeding  from  the  anode  that  reach 
through  the  grid  represent  the  stray  field  due  to  Es,  which  therefore 
tends  to  draw  the  electrons  through  the  grid  and  throw  them  on  to 
the  anode.  By  varying  the  potential  Eg  of  the  grid  the  intensity 
of  the  field  between  the  grid  and  cathode  is  varied  in  such  a  manner 
that  the  effect  of  Eg  is  similar  to  that  of  Es,  and  whether  or  not 
electrons  will  flow  away  from  the  cathode  depends  on  the  resultant 
value  of  Es  and  Eg.  Now,  Es  is  always  positive  and  therefore 
E,+Eg  will  be  positive  (1)  when  Eg  is  positive,  and  (2)  if  Eg  is  nega- 
tive and  less  than  E,. ' 

(1)  When  Eg  is  positive  some  of  the  electrons  moving  away 


THE  THERMIONIC  AMPLIFIER 


149 


from  the  cathode  are  drawn  to  the  grid  (see  Fig.  69),  while  the 
rest  are  drawn  through  the  openings  of  the  grid  to  the  anode  under 
the  influence  of  Es.  The  relative  number  of  electrons  going  to 
and  through  the  grid  depends  upon  the  mesh  of  the  grid,  the 
diameter  of  the  grid  wires  and  the  relative  values  of  Es  and  Ea. 
When,  for  example  Es  is  large  compared  with  Eg  the  number  of 
electrons  going  to  the  grid  is  comparatively  small,  but  for  any 
fixed  value  of  E,  the  grid  current  increases  rapidly  with  increase 
in  Eg.  Hence,  for  positive  values  of  Eg  current  will  be  established 
in  the  grid  circuit  FGEg  (Fig.  72). 

(2)  If,  however,  Eg  is  negative  and  less  than  Es,  as  is  generally 
the  case,  nearly  all  the  electrons  drawn  away  from  the  cathode  pass 
to  the  plate,  practically  none  going  to  the  grid.  In  this  case  the 
resistance  of  the  grid  circuit  is  practically  infinite  for  low  frequen- 


FIG.  72. 

cies.  The  electrostatic  capacities  between  the  electrodes  causes  the 
impedance  between  filament  and  grid  to  have  a  value  depending  on 
the  output  circuit.  For  the  present  we  shall  neglect  this  effect, 
which  is  usually  small,  and  later  on  investigate  the  conditions 
under  which  it  can  manifest  itself  to  a  marked  extent. 

If,  now,  an  alternating  E.M.F.  be  impressed  on  the  grid  circuit 
so  that  the  grid  becomes  alternately  positive  and  negative  with 
respect  to  the  filament  or  cathode,  the  resistance  of  the  grid 
circuit  FGEg  which  is  usually  referred  to  as  the  input  circuit, 
will,  if  the  frequency  is  not  too  high,  be  practically  infinite  for  the 
half  cycle  that  the  grid  is  negative  and  finite  but  variable  for  the 
positive  half  cycle.  If,  on  the  other  hand,  the  alternating  E.M.F. 
be  superimposed  upon  a  constant  negative  grid  potential,  which 
is  so  chosen  with  respect  to  the  value  of  the  impressed  alternating 
voltage  that  the  grid  always  remains  negative  with  respect  to  the 
filament,  the  resistance  of  the  input  circuit  is  infinite. 


150  THERMIONIC  VACUUM  TUBE 

It  can  now  be  seen  in  a  general  way  how  the  device  functions 
as  a  relay.  Any  variation  in  the  grid  potential  changes  the  in- 
tensity of  the  field  between  filament  and  grid,  resulting  in  a  corre- 
sponding change  in  the  number  of  electrons  moving  from  fila- 
ment to  plate.  Hence  potential  variations  set  up  between  filament 
and  grid  cause  variations  in  current  in  the  output  circuit  PEbr0, 
the  power  developed  in  the  load  TQ  being  greater  than  that  expended 
in  the  input  circuit. 

53.  Current-voltage  Characteristics  of  the  Thermionic  Ampli- 
fier. Returning  now  to  a  consideration  of  the  expression  for 
the  current 


(2) 


it  is  to  be  noticed  in  the  first  place  that  since  this  equation  contains 
two  independent  variables,  Ep  and  Eff,  the  three-electrode  device 
possesses  two  families  of  characteristics,  or  the  complete  charac- 
teristic can  be  represented  by  a  surface.  The  current  as  a  function 
of  the  filament-plate  voltage  Ep  can  for  various  negative  values 
of  filament-grid  voltage  Eg  be  represented  by  a  series  of  curves 
such  as  those  shown  in  Fig.  73.  It  will  be  noticed  that  each 
of  these  curves  is  similar  to  the  current-voltage  characteristic  of 
the  simple  two-electrode  thermionic  valve  discussed  in  Chapter  IV. 
The  main  difference  is  that  in  the  three-electrode  tube  the  current 
is  limited  not  only  by  space  charge  and  the  voltage  drop  in  the 
filament,  but  also  by  the  grid.  For  the  same  potential  on  the 
plate  the  current  in  the  three-electrode  tube  will  therefore  be 
smaller  than  in  a  simple  valve.  This  follows  directly  from 
quation  (2). 

The  relation  between  Ip  and  Eff  for  various  values  of  Ep  can  be 
expressed  by  a  set  of  curves  similar  to  those  shown  in  Fig.  73. 
Fig.  74  shows  such  a  set  of  characteristics.  The  ordinates  repre- 
sent current  to  the  plate  and  not  necessarily  the  emission  current, 
i.e.,  total  current  from  the  filament.  When  the  grid  becomes  posi- 
tive it  takes  current  and  so  can  distort  the  Ip,  Eg  curves.  A  set 
of  grid  current  curves  for  various  plate  potentials  is  shown  in  Fig. 
75.  For  the  higher  plate  potentials  these  curves  show  a  maximum. 
This  is  due  to  secondary  electron  emission  from  the  grid  by  the 
impact  of  electrons  coming  from  the  filament  (see  p.  47)  .  At  the 

E 
lower  plate  potentials  the  stray  field  given  by  —  is  smaller,  and 


THE  THERMIONIC  AMPLIFIER 


151 


fewer  electrons  are  attracted  to  the  plate.  The  secondary  electron 
emission  is  then  also  less  marked,  so  that  the  Ig—Eg  curve  shows  a 
rapid  increase  of  Ig  with  increasing  *Eg. 

So  far  it  has  not  been  possible  to  derive  the  equation  of  the 
whole  characteristic  theoretically  with  sufficient  accuracy.     For 


150 
Anode  Volts 

FIG.  73. 


the  operating  range  of  the  characteristic,  when  the  tube  functions 
as  an  amplifier,  the  plate  current  can  be  expressed  by  the  equation.1 


(3) 


where    2EP  and   2Eg  are   the  filament-plate  and  filament-grid 
1  H.  J.  VAN  DER  BIJL,  Phys.  Rev.,  Vol.  12,  p.  180,  1918. 


152 


THERMIONIC  VACUUM  TUBE 


voltages.      If,  for   example,    an  alternating   e.m.f.  e  sin   pt   be 
impressed  on  the  grid  circuit  the  equation  takes  the  form: 


sn 


(4) 


Grid  Voltb. 

FIG.  74 

This  equation  was  determined  empirically  and  is  subject  to 
certain  limitations.  In  the  first  place,  it  does  not  apply  to  the 
horizontal  part  of  the  characteristic  which  gives  the  saturation 
current,  but  only  to  that  part  which  obtains  when  the  filament  is 
hot  enough  to  emit  more  electrons  than  are  needed  for  the  current 
convection  through  the  tube.  This  is  the  condition  under  which 
the  amplifier  operates,  because  here  the  plate  current  can  be 


THE  THERMIONIC  AMPLIFIER 


153 


varied  by  varying  any  of  the  applied  voltages.  Another  condition 
for  equation  (4)  is  that  the  grid  should  not  become  sufficiently 
positive  to  distort  the  characteristic.  Under  these  conditions  I 
have  generally  found  this  equation  to  hold  sufficiently  well,  to  a 
first  approximation  at  least,  and  have  been  using  it  in  connection 
with  work  on  the  amplifier  tube.  The  above  equation  does,  how- 
ever, not  hold  sufficiently  accurately  for  purposes  of  radio  detection, 
since  this  is  determined  by  second  order  quantities. 


15  20  25  30  35  40 


0.4 


FIG.  75. 


Latour  1  has  derived  some  equations  for  the  "  relay  effect " 
of  audion  tubes.  He  starts  from  the  general  functional  expressions 
for  the  plate  and  grid  currents:  If=F(Ep}  Eg)  and  Ig=f(Eg,  Ep). 
In  the  expansion  of  these  equations  he  neglects  all  quantities  of 
the  second  and  higher  order,  thus  assuming  that  the  current  and 
voltage  variations  are  very  small,  or  that  the  characteristic  is 
linear  over  the  operating  range. 

1  M.  LATOUR,  Electrician,  December,  1916. 


154  THERMIONIC  VACUUM  TUBE 

Vallauri 1  also  assumes  a  linear  characteristic  by  expressing 
the  equation  for  the  plate  current  in  the  form: 

Ip  =  aE0+bEP+c. 

It  will  be  shown  later  that  it  is  important  to  distinguish  between 
the  characteristic  of  the  tube  itself  and  that  of  the  tube  and  exter- 
nal circuit  combined.  The  latter  can  by  taking  special  precau- 
tions be  made  practically  linear.  The  characteristic  of  the  tube 
itself  can,  however,  not  be  regarded  as  linear  over  the  range  over 
which  most  amplifiers  operate.  The  curvature  of  the  characteris- 
tic cannot  be  neglected  because  it  introduces  distortion  which, 
unless  properly  taken  care  of,  makes  it  practically  worthless  as  a 
telephone  repeater,  for  example,  on  long  telephone  lines.  When 
treating  the  tube  as  an  oscillation  generator  the  curvature  of  the 
characteristic  can  be  neglected,  because  the  oscillation  current  is 
established  in  an  oscillation  circuit  which  is  usually  tuned  suf- 
ficiently sharply  to  eliminate  the  harmonics  caused  by  the  curva- 
ture of  the  characteristic. 

Equation  (3)  gives  the  characteristic  of  the  tube  itself;  that 
is,  Ep  and  Eg  are  the  potentials  of  the  plate  and  grid  with  respect 
to  the  filament,  and  are  not  necessarily  equal  to  the  plate  and  grid 
battery  voltages.  Ep  is,  for  example,  only  equal  to  the  plate 
battery  voltage  Eb  when  the  external  resistance  TQ  is  zero  (Fig.  72) . 
When  TO  is  not  zero  the  potential  drop  established  in  ro  by  the  cur- 
rent in  the  plate  circuit  causes  a  decrease  in  Ep,  and  it  can  readily 
be  seen  that  if  the  current  be  varied,  by  varying  the  grid  potential, 
Ep  becomes  a  function  of  the  plate  current.  This  effect  will  be 
discussed  more  fully  when  we  come  to  consider  the  characteristic 
of  the  tube  and  circuit  (Section  58). 

Langmuir2  has  expressed  the  equation  for  the  characteristic 
as 

Ip=A(Ep+kE0Y/2. 

The  extent  to  which  the  characteristics  of  practical  tubes 
depart  from  the  f-power  relation  was  discussed  in  Chapter  IV. 
In  the  case  of  the  two-electrode  tube  the  main  cause  of  the  devia- 
tion is  the  voltage  drop  in  the  filament.  This  has  a  greater 

1  G.  VALLAURI,  L'Elettrotecnica,  Vol.  4,  1917,  Electrician,  Vol.  80,  p.  470, 
1917. 

2  I.  LANGMUIR,  Proc.  I.R.E.,  p.  278,  1915. 


THE  THERMIONIC  AMPLIFIER  155 

effect  at  the  lower  than  at  the  higher  voltages.  In  three-electrode 
tubes  the  limitation  of  current  by  the  grid  accentuates  this  devia- 
tion. Thus,  referring  to  equation  (3),  the  constant  ;u  is  generally 
greater  than  unity  and,  therefore,  although  the  plate  voltage 

(E  \ 

— +.#0+  e)  is  low,  so  that 

the  voltage  drop  in  the  filament  has  a  relatively  greater  effect 
in  causing  a  deviation  from  the  | -power  relation.  As  an  example, 
suppose  that  £^=100  volts;  M  =  5;  the  voltage  in  the  filament 
E/=lQj  and  Eg+e  =  Q.  Then  the  effective  voltage  is  only  twice 
the  voltage  drop  in  the  filament.  Under  such  conditions  the 
deviation  from  the  f -power  relation  is  considerable.  It  is  for  this 
reason  that  the  quadratic  equation  (3)  is  generally  found  to  be  more 
serviceable  at  least  for  that  range  of  the  characteristic  over  which 
the  tube  operates  as  an  amplifier. 

The  quantity  e,  which  depends  on  the  intrinsic  potential 
difference  between  the  filament  and  the  system  constituting  the  grid 
and  plate,  is  usually  small,  but  may,  in  some  types  of  tubes,  vary 
considerably.  For  tubes  operating  with  high  effective  voltages  e 
can  generally  be  neglected.  But  when  the  effective  voltage  is  low, 
as  in  the  detector  and  small  amplifier  tubes,  variations  in  e  can, 
if  not  corrected  for,  cause  deviations  in  the  exponent  of  the  effect- 
ive voltage. 

The  important  thing  about  the  tube  equation  is  that  the  cur- 
rent can  be  expressed  as  a  function  of  (Ep+^Eg)1. 

Referring  to  equation  (3)  and  Fig.  74,  we  see  that  the  current 
is  finite  for  negative  values  of  the  grid  potential,  and  is  reduced  to 
zero  only  when 


H- 


Eg=-  p?+e  )=ES (5) 


This  linear  relation  and  equation  (3)  can  be  verified  experi- 
mentally when  the  constants  ^  and  e  are  known.  These  constants 

1  This  expression  for  the  effective  voltage  in  a  three-electrode  tube  was 
established  experimentally  by  the  author  and  published  in  1913  (Verh.  d.  D. 
Phys.  Gesell.,  Vol.  15,  p.  330,  1913).  See  also  p.  44.  The  same  expression 
has  also  been  used  by  SCHOTTKY  (Archiv.  f.  Elektrotechnik,  Vol.  8,  p.  1,  1919. 
BARKHAUSEN  (Jahrb.  d.  drahtlosen  Tel.  &  Tel.,  Vol.  14,  p.  27,  1919)  and 
others.  See  also  W.  H.  ECCLES  (Rad.  Rev.,  Vol.  1,  p.  69,  Nov.,  1919). 


156  THERMIONIC  VACUUM  TUBE 

can  be  determined  by  methods  which  do  not  involve  the  exponent 
of  equation  (3).     Let  us  assume  a  general  exponent  /3,  thus: 


Assuming  the  general  case  in  which  both  Ep  and  E0  are  variable, 
we  have: 

dlp=3lp  dEp    dlp 
dEg     SEpdEffdEg' 
Now 


Hence 

/IT  /w.  \  0-i/f  ,77?        \ 

-_     (6) 


Since  the  current  can  be  varied  by  varying  either  one  or  both 
of  the  independent  variables  Ep  and  Eg,  we  can  make  these  varia- 
tions in  accordance  with  the  condition  that  the  current  Ip  remains 
constant;  for  example,  the  current  can  be  first  increased  by 
increasing  Ep  and  then  brought  back  to  its  original  value  by 
increasing  the  negative  grid  voltage  Eg.  The  relation  between 
the  variations  in  Ep  and  E0  necessary  to  keep  the  current  con- 
stant, can  be  obtained  by  putting  Ip=  constant  in  equation  (6). 
Then  we  have  either 


€  =  0  .     .......    (5a) 

or 


These  equations  are  therefore  independent  of  the  exponent  of 
(3).  Equation  (5a)  obviously  states  the  condition  that  the  current 
has  the  constant  value  zero,  and  shows  that  the  stray  field  poten- 
tial Es  is  simply  equal  to  the  absolute  value  of  the  grid  potential 
which  is  necessary  to  reduce  the  plate  current  to  zero. 

Referring  to  the  above  equations  for  the  partial  derivatives  of 
IP,  it  follows  that  a  change  iii  the  grid  potential  produces  ju-times  as 


THE  THERMIONIC  AMPLIFIER 


157 


great  a  change  in  the  plate  current  as  an  equal  change  in  the  plate 
voltage. 

Equation  (7)  can  be  interpreted  to  mean  that  a  potential  varia- 
tion 5Eg  =  eg  impressed  between  the  grid  and  the  filament  is  equiv- 
alent to  introducing  an  E.M.F.  in  the  plate  circuit  which  is  equal 
to  fjLeg. 

This  result  is  of  fundamental  importance  and  has  been  found  of 
great  value  in  the  solution  of  many  vacuum  tube  problems. 


1600, 


1400 


«J  1200 


1000 


1-8 


ZA- 


Grid 
FIG.  76. 

Integrating  equation  (7)  we  get 

E'P  =  Ep+nEg  ........     (8) 

While  equation  (5a)  gives  the  relation  between  Ep  and  Eg  necessary 
to  neutralize  the  stray  field  and  keep  the  current  zero,  equation  (7) 
gives  the  relation  necessary  to  keep  the  current  constant  at  any 
convenient  value.  The  verification  of  these  relations  is  shown 
in  Fig.  76.  l  The  slope  of  these  curves  is  equal  to  the  constant  /z. 

1  H.  J.  VAN  DER  BIJL,  Phys.  Rev.,  Vol.  12,  p.  171,  1918.     See  also  Fig.  14, 
p.  45. 


158 


THERMIONIC  VACUUM  TUBE 


The  characteristic  equation  (3)  was  verified  as  follows:  The 
tube  was  inserted  in  a  circuit  such  as  shown  in  Fig.  72,  with  the 
exception  that  the  generator  in  the  input  circuit  and  the  resist- 
ance ro  were  omitted.  A  convenient  negative  potential  was  applied 
to  the  grid,  so  that  no  current  could  be  established  in  the  grid 


600 


800 


circuit,  and  the  current  in  the  plate  circuit  observed  as  a  function 
of  the  plate  voltage  Ep.  Since  ro  was  zero.Ep  was  always  equal 
to  Eb)  the  plate  battery  voltage.  The  grid  being  kept  at  a  constant 

negative  potential  Eg  with  respect  to  the  filament,  current  could 

•pi 

not  be  established  in  the  plate  circuit  until  the  —  +  e  became 

M 

greater  than  E0.    The  characteristic  obtained  is  shown  in  Fig.  77. 


THE  THERMIONIC  AMPLIFIER 


159 


From  the  value  of  the  plate  voltage  for  which  the  current  is  just 
reduced  to  zero  we  get 


E4 

,5 

I 

V 

•o 


40 


80 


160 


IQQ 


240 


280 


FIG.  78. 


and  since  &  could  be  determined  as  explained  above,  this  equation 
could  be  used  to  give  e.  Once  /z  and  c  are  known  the  observed 
current  can  be  plotted  as  a  function  of  the  expression 


for  arbitrary  values  of  Ep  or  Eg.    Some  curves  obtained  in  this  way 
are  shown  in  Fig.  78. 

If  we  obtain  a  number  of  characteristics  such  as  those  shown 


160  THERMIONIC  VACUUM  TUBE 

in  Fig.  74,  which  show  the  relations  between  the  plate  current  and 
grid  potential  for  a  number  of  different  plate  potentials  and  plot 
the  logarithms  of  Ip  against  the  logarithms  of  the  effective  voltage 

(— -f- Eg]  the  observed  points  for  all  the  characteristics  should, 

x  M 

according  to  equation  (3),  lie  on  one  straight  line.     This  can  be 

done  by  subtracting  the  applied  grid  potentials  from  the  grid 
potential  which  is  just  necessary  to  reduce  the  current  to  zero, 
and  plotting  on  logarithmic  paper  the  values  so  obtained  against 

the  observed  currents.     (Note  that  the  value  of  the  grid  potential 

•pi 

necessary  to  reduce  the  current  to  zero  is  — .)     The  disadvantage 

of  such  a  procedure  lies  in  the  uncertainty  of  the  voltage  at  which 
the  current  becomes  zero.  However,  the  logarithmic  plot  of  the 
curves  of  Fig.  74,  and  which  is  shown  in  Fig.  79  indicates  a  substan- 
tially good  verification  of  equation  (3).  The  slope  of  this  lumped 
logarithmic  line  is  almost  exactly  2. 

54.  Amplification  Constant.    The  constant  ju  appearing  in  the 
above  equations  is  one  of  the  most  important  constants  of  the 
audion  or  three-electrode  tube.     It  will  be  shown  later  that  /-c  is  the 
maximum  voltage  amplification  obtainable  from  the  tube.     This 
constant  is  also  very  instrumental  in  determining  the  current  and 
power  amplification  and  can  therefore  be  referred  to  as  the  ampli- 
fication constant.     This  constant  plays  an  important  part  in  all 
functions  of  the  tube,  as  will  be  shown  later  when  we  come  to 
consider  its  use  as  a  radio  detector,  modulator,  oscillation  gen- 
erator, etc.     It  will  be  noticed  that  since  it  appears  in  the  stray 
field  relation  (equation  (1)),  which  is  a  pure  potential  relation, 
the  amplification  constant  is  a  function  only  of  the  geometry  of 
the  tube.     It  depends,  for  example,  on  the  mesh  of  the  grid, 
diameter  of  the  grid  wire  and  the  distance  between  grid  and  plate. 
It  can  be  determined  from  EPE0 — curves  shown  in  Fig.  76  and  by 
methods  which  will  be  described  later.     In  practice  it  is  generally 
found  that  //  is  not  quite  constant,  its  value  decreasing  somewhat 
at  lower  voltages.     For  the  operating  range  of  voltages  commonly 
employed  its  value  does,  however,  not  vary  much.     (See  Fig.  125.) 

55.  Plate  Resistance  and  Impedance.    The  resistance  of  a  tube 
is  due  to  the  work  which  the  electrons  emitted  from  the  cathode 
must  do  in  moving  from  cathode  to  anode.    Let  us  consider  the 
case  of  a  single  electron  emitted  from  the  cathode.     In  moving 


THE  THERMIONIC  AMPLIFIER 


161 


through  the  cathode  surface  it  has  to  do  an  amount  of  work 
equivalent  to  the  electron  affinity  and  in  moving  from  cathode 
to  anode  it  has  to  do  work  in  overcoming  the  contact  potential 
difference  between  cathode  and  anode.  This  may  sometimes 
assist  the  electron  in  moving  from  cathode  to  anode.  (See  Chap- 
ter III.)  The  total  amount  of  work  it  has  to  do  to  overcome  these 


o 


rcs 

CD 


^ 


± 


-K 


t 


* 


- 


ZO 


30 


40 


FIG.  79. 

forces  is  generally  small  and  never  amounts  to  more  than  a  drop  of 
a  few  volts.  If  these  were  the  only  forces  exerted  on  a  large  num- 
ber of  electrons  escaping  from  the  cathode  the  application  of  a  small 
voltage  between  cathode  and  anode  would  almost  immediately 
give  rise  to  the  saturation  current,  and  the  resistance  of  the  tube 
would  for  all  values  of  current  less  than  the  saturation  current  be 
very  low.  This  is,  however,  not  the  case,  since  the  electrons 


162 


THERMIONIC  VACUUM  TUBE 


in  the  space  exert  a  mutual  repelling  force  on  one  another.  This 
is  the  space  charge  effect  explained  in  Chapters  I  and  IV,  and 
causes  by  far  the  greatest  expenditure  of  energy  on  the  part  of  the 
electrons  in  moving  to  the  anode.  This  expenditure  of  energy 
causes  the  heating  of  the  anode. 

The  true  d-c.  resistance  of  the  tube  is,  of  course,  given  simply 
by  the  ratio  of  the  total  amount  of  work  done  to  the  square  of 

Tjl 

the  current,  i.e.,  by  j^.    The  a-c.  resistance  on  the  other  hand,  is 
IP 

given  by  the  slope  of  the  plate  current  characteristic,  and  since  the 
characteristic  is  non-linear  the  a-c.  and  d-c.  resistances  are  not  the 
same.  Referring  to  Fig.  80,  the  d-c.  resistance  at  a  voltage  Ep  is 


FIG.  80. 


given  by  the  reciprocal  of  the  slope  of  the  straight  line  OC,  while  the 
impedance  of  the  tube  is  given  by  the  ratio  of  the  alternating 
voltage  ep  between  filament  and  plate  to  the  alternating  current 
ip  in  the  plate  circuit.  Now,  the  flow  of  electrons  in  the  tube 
shows  no  lag,  and  for  frequencies  low  enough  to  make  the  effect 
of  the  electrostatic  capacity  of  the  tube  itself  negligibly  small, 
the  condensive  reactance  thus  being  also  practically  infinite,  the 

/>  /^/7 

impedance  is  simply  given  by  -^=-r  (see  Fig.  80),  and  is  then 

lp      CLO 

of  the  nature  of  a  pure  resistance.  For  most  tubes  used  at  present 
this  approximation  is  satisfactory  for  frequencies  up  to  the  order  of 
several  hundred  thousand  cycles  per  second.  For  a  tube  like  that 
shown  in  Fig.  68,  for  example  the  filament-plate  capacity  is  of 


THE  THERMIONIC  AMPLIFIER 


163 


the  order  of  a  few  micro-microfarads.     Now  we  have  —  —  ^-~ 

ep     dEP 

when  ep  and  ip  are  very  small.  But  in  practice  we  generally  do  not 
deal  with  very  small  current  variations.  To  obtain  an  expression 
for  a-c.  resistance  for  finite  variations  we  must  evaluate  the  partial 

derivative  —^-  from  the  equation  of  the  characteristic  and  integrate 
it  over  a  complete  cycle  of  variations,  thus: 


,„, 


For  frequencies  at  which  the  electrostatic  capacity  of  the 
tube  cannot  be  regarded  as  negligibly  small,  we  have  in  effect  a 
condenser  in  shunt  with  the  tube  resistance.  If  x  is  the  reactance 
due  to  the  capacity  of  the  tube  the  plate  impedance  Zp  can  be 
obtained  from  the  admittance  Yp: 


where 


p 


rf+4 


(10) 


(11) 


To  evaluate  expression  (9)  let  us  assume  a  general  exponent 
for  the  characteristic  equation: 


Then 


where 


Hence 


.       .    .     .     (12) 


oinEyn- 


2-jriJL 


1  r<i*l         f> 

-j,  (1+isin^ 


n-l 


Now  the  maximum  value  e  of  the  input  voltage  is  never 
greater  than  Ey;    for  distortionless  amplification  e  must  always 


164 


THERMIONIC  VACUUM  TUBE 


be  less  than  Ey  (see  Section  60).  Referring,  for  example,  to  Fig. 
81,  it  will  be  seen  that  Ey  is  the  intercept  cd  when  Eg  =  0  or 
fd  when  Eg  =  cf.  Taking  the  latter  case  it  will  be  seen  that  the 
maximum  value  of  the  input  voltage  e  should  not  exceed  the  value 
fd  otherwise  we  would  be  working  beyond  the  point  d,  and  then 
the  lower  peaks  of  the  output  current  wave  would  be  chopped  off 


Grid  Voltage 
FIG.  81. 


thus  introducing  harmonics.  Furthermore,  since  the  maximum 
value  of  sin  pt  is  unity  and  its  odd  powers  vanish  on  integration 
the  expression  in  the  parentheses  can  be  expanded  into  a  series, 
the  integral  of  which  converges  sufficiently  rapidly  to  enable  us  to 
compute  the  resistance,  for  all  practical  values  of  n,  from  a  few 
terms  of  the  expansion.  The  integrated  series  is: 


THE  THERMIONIC  AMPLIFIER  165 

(n-1)   .  .  .  (n-4)/  e  \4 


1  _<mff/-lr       (n-l)(n-2)/  e  \2 

FSK       M       L  ~  (2)2         iFj  " 


(2.4) 

(n-1)   .  .  .  (n- 
-~ 


For  the  case  of  the  amplifier  w  =  2,  and  all  the  terms  except 
the  first  vanish  so  that  we  get: 


The  a-c.  plate  resistance  is  therefore  to  a  first  approximation 
independent  of  the  a-c.  input  voltage. 

It  is  customary  to  speak  generally  of  the  impedance  of  the  tube, 
meaning  thereby  the  plate  impedance.  It  must,  however,  be 
remembered  that  unless  the  frequency  is  very  high,  the  wattless 
component  of  the  impedance  is  practically  zero,  and  the  impedance 
is  then  given  by  equation  (14)  in  which  case  it  is  of  the  nature  of  a 
pure  resistance.  For  a  parabolic  characterictic  (w  =  2)  equation 
(14)  is  simply  the  slope  of  the  IP—EP  characteristic. 

56.  Mutual  Conductance.  So  far  we  have  considered  only  the 
plate  voltage-plate  current  characteristic  from  which  we  have 
deduced  the  plate  impedance  by  obtaining  an  expression  for  the 
variation  in  plate  current  as  a  function  of  the  plate  potential 
variations.  The  usual  thing  in  practice  is  to  vary  the  plate  current 
by  varying  the  grid  potential.  As  will  be  seen  from  the  following 
the  effects  in  this  case  can  be  deduced  directly  from  the  previous 
considerations  by  the  introduction  of  the  amplification  constant  /*. 

Referring  to  the  fundamental  equation  for  the  characteristic, 
we  have 


and 


Equation  (16)  gives  the  slope  of  the  plate  current-grid  potential 
characteristic  and  (17)  gives  the  mutual  conductance  *  gm.  It 
will  be  noticed  that 


(18) 


1  The  expression  "  mutual  conductance  "  for  this  quantity  was  suggested 
by  HAZELTINE  (Proc.  I.  R.  E.,  Vol.  6,  p.  63,  1918). 


166 


THERMIONIC  VACUUM  TUBE 


This  is  a  very  important  quantity  and  is  involved,  as  will  be  shown 
later,  in  all  expressions  giving  the  degree  of  merit  of  the  tube  when 
functioning  as  amplifier,  detector,  oscillator,  etc.  It  is  always 
desirable  to  have  the  mutual  conductance  as  large  as  possible. 
While  n  depends  almost  entirely  on  the  structure  of  the  grid  and 
its  position  relative  to  tile  other  electrodes,  rp  depends  upon  /* 
and  the  the  surface  areas  of  cathode  and  anode  as  well. 

The  mutual  conductance  gives  a  measure  for  the  effect  of  the 
grid  potential  on  the  plate  current.  The  analogous  expression  for 
the  effect  of  the  plate  potential  on  the  grid  current  is  given  by 


(17a) 


and  may  be  called  the  reflex  mutual  conductance.  At  frequencies 
for  which  electrode  capacities  are  effective  to  an  appreciable  extent 
we  have  to  consider  the  mutual  impedances  Zm  =  gm+jxm,  etc., 
which  cannot  be  obtained  from  the  static  characteristics  of  the 
tube. 

57.  Shape  of  Output  Wave  in  Circuit  of  Low  External  Im- 
pedance.   Consider  the  case  of  the  tube  circuit  shown  in  Fig.  82 


FIG.  82. 

and  let  a  voltage  e  sin  pt  be  impressed  between  filament  and  grid. 
The  resistance  n  may  be  that  of  the  input  transformer  coil  which 
is  supposed  to  be  wound  to  work  into  a  practically  open  circuit. 
For  the  present  we  shall  suppose  that  the  external  impedance  ZQ 
in  the  output  is  negligibly  small  compared  with  the  plate-resistance 
of  the  tube,  so  that  the  characteristic  of  the  circuit  is  practically 
the  same  as  that  of  the  tube. 

If  the  constant  grid  voltage  Ec  is  sufficiently  negative  to  insure 
that  the  grid  never  takes  current,  the  wave  shape  in  ZQ  is  deter- 


THE  THERMIONIC  AMPLIFIER 


167 


mined  by  the  characteristic  equation  (4).  Thus  if  the  voltage  is 
a  sinusoid  (Fig.  83,  curve  a)  the  output  current  is  a  lop-sided  curve 
shown  in  Fig.  83,  curve  W.  This  can  readily  be  seen  by  referring 
to  Fig,  81,  from  which  it  will  be  seen  that  if  the  potential  of  the 


for 


b1 


grid  be  varied  about  the  value  Ec=cf,  the  increase  ab  in  plate  cur- 
rent due  to  the  decrease  oa  in  the  negative  grid  potential  is  greater 
than  the  decrease  a'b'  in  current  caused  by  an  equal  increase 
oa'  in  the  negative  grid  potential.  This  would  produce  distortion 


168  THERMIONIC  VACUUM  TUBE 

since  the  output  current  is  not  an  exact  reproduction  of  the  input. 
The  curve  b  of  Fig.  83,  of  course,  shows  the  variation  in  the  output 
current  from  its  mean  d-c.  value  which  obtains  when  the  input 
alternating  voltage  is  zero. 

Now,  suppose  the  grid  battery  be  so  adjusted  that  whenever 
e  sin  pt  is  positive  the  grid,  takes  current.  Since  the  grid  current 
characteristic  is  of  the  nature  shown  in  Fig.  75,  the  grid  current 
wave  will  be  given  by  d  (Fig.  83),  providing  the  grid  potential  does 
not  become  sufficiently  high  to  cause  the  emission  of  secondary 
electrons  from  it.  Now  this  current  in  the  grid  circuit  causes  a 
voltage  drop  in  rt  thus  lowering  the  potential  difference  between 
filament  and  grid.  The  output  current  wave  may  therefore  take 
the  shape  shown  by  curve  cb'.  This  would  therefore  minimize 
the  distortion  if  the  quantities  involved  were  correctly  propor- 
tioned. On  the  other  hand,  once  the  grid  current  is  established 
it  increases  so  rapidly  with  further  increase  in  the  grid  voltage 
that  the  increase  in  plate  current  during  the  half  cycle. when  the 
grid  is  positive  can  become  less  than  the  decrease  during  the  other 
half  cycle,  and  this  materially  lowers  the  amplification. 

Referring  again  to  the  case  in  which  the  grid  is  kept  negative 
with  respect  to  the  filament,  it  will  be  seen  on  expanding  equation 
(4)  that  the  curve  bb'  (Fig.  83)  consists  of  the  following  components: 


T  (19) 


The  first  term  represents  the  steady  direct  current  in  the  plate  cir- 
cuit which  is  maintained  when  the  input  e  is  zero,  and  is  the  value 
about  which  the  plate  current  varies  for  finite  values  of  e.  The 
second  term  gives  the  alternating  output  current  (ee,  Fig.  83) 
which  is  in  phase  with  the  input  voltage  and  which  is  the  only 
useful  current  for  amplification  purposes.  The  harmonic  repre- 
sented by  the  third  term  and  having  double  the  frequency  of  the 
fundamental  is  present,  as  was  to  be  expected  from  the  parabolic 
shape  of  the  characteristic.  It  is  shown  by  the  curve  ff  in  Fig.  83. 
This  is  the  undesirable  term  which  causes  distortion.  The  last 
term,  which  is  proportional  to  the  square  of  the  input  voltage 
is  the  change  in  the  d-c.  component  due  to  the  alternating  input 


THE  THERMIONIC  AMPLIFIER  169 

voltage  (shown  by  the  broken  line  in  Fig.  83),  and  is  the  only  effect- 
ive component  of  the  output  current  when  using  the  device  as  a 
radio  detector. 

If  the  output  transformer  To  (Fig.  82)  has  a  sharp  frequency 
characteristic,  and  a  pure  note  be  impressed  on  the  input  of  the 
tube,  a  current  meter  AI  inserted  in  the  load  circuit  would  indicate 
a  current  which  is  proportional  only  to  the  second  term  of  equation 
(19),  that  is,  to  the  fundamental,  and  the  distortion  produced 
by  the  curvature  of  the  characteristic  would  not  be  a  serious  matter. 
In  telephony  we  have  to  deal,  however,  with  frequencies  ranging  up 
to  about  3000  cycles  per  second,  and  it  is  desirable  to  use  a  trans- 
former with  a  flat  frequency  characteristic,  so  that  all  frequencies 
are  transmitted  with  more  or  less  equal  facility.  In  such  case  the 
harmonic  term  would  cause  serious  distortion  of  the  speech 
wave.  Distortion  can,  however,  be  reduced  to  a  negligible  quan- 
tity by  properly  choosing  the  impedance  of  the  output  transformer. 
We  have  so  far  assumed  that  the  transformer  impedance  in  the 
direction  Z$Z\  is  small  compared  with  the  plate  resistance  of  the 
tube.  In  practice  this  is  not  so.  In  fact  the  best  operation  as 
power  amplifier  is  obtained  when  this  impedance  is  approxi- 
mately equal  to  the  plate  resistance.  In  such  case  the  character- 
istic of  the  circuit  is  different  from  that  of  the  tube  alone,  as  we 
shall  now  proceed  to  show. 

58.  Characteristic  of  Circuit  Containing  Tube  and  Resistance 
in  Series.  Let  us  first  consider  the  simple  case  in  which  the  out- 
put circuit  is  non-reactive  and  has  a  resistance  ro  (Fig.  72).  Let 
the  plate  current  be  measured  for  different  values  of  grid  potential 
Eg,  the  plate  battery  voltage  Eb  remaining  constant.  When  the 
grid  is  so  much  negative  that  the  current  in  the  plate  circuit  is 
reduced  to  zero,  the  plate  voltage  Ep  is  equal  to  the  plate  battery 
voltage  Ei).  But  when  current  is  established  in  the  plate  circuit 
there  is  a  voltage  drop  across  ro,  and  Ep  will  be  less  than  Eb,  being 
given  by 

EP  =  Eb-r0Ip, .     (20) 

where  Ip  is  the  plate  current,  Ep  thus  becomes  a  variable.  Sub- 
stituting this  value  of  EP  in  the  characteristic  equation,  thus: 


170  THERMIONIC  VACUUM  TUBE 

we  get,  putting 


This  is  the  equation  for  the  characteristic  of  the  circuit  consisting 
of  the  tube  and  resistance  ro,  and  it  will  be  seen,  if  Ip  be  plotted 
against  E'y,  for  various  values  of  ro,  that  the  curvature  of  the 
characteristic  is  reduced  as  ro  is  increased,  the  characteristic 
becoming  practically  linear  when  ro  is  equal  to  or  greater  than  the 
plate  resistance.  This  is  an  important  result  for  which  I  am 
indebted  to  my  associate  Dr.  H.  D.  Arnold,  and  has  an  important 
bearing  on  the  problem  of  distortionless  amplification  of  telephonic 
currents.1 

The  effect  of  the  resistance  on  the  characteristic  of  the  output 
circuit  is  shown  graphically  in  Figs.  84  and  85.  In  the  first  the 
plate  battery  voltage  Eb  had  a  constant  value  equal  to  n(E' 0-\- e) , 
where  E'tt  is  given  by  00',  while  in  Fig.  85  the  plate  battery  was 
so  adjusted  for  eveiy  value  of  ro  as  to  keep  Ep  constant  for  zero 
grid  voltage.  It  will  be  noticed  that  when  ro=rp,  the  character- 
istic is  substantially  linear  over  a  considerable  range  of  input 
voltage. 

59.  Static  and  Dynamic  Characteristics.  So  far  we  have  con- 
sidered only  the  static  characteristics  of  the  tube  and  its  circuit. 
We  have  seen  that  the  static  characteristics  of  the  tube  itself, 
that  is,  the  characteristics  which  are  obtained  when  the  external 
resistance  is  neglibibly  small  in  comparison  with  the  plate  resist- 
ance of  the  tube,  are  given  by  equation  (3),  while  those  of  the 
output  circuit,  containing  an  external  resistance  as  well  as  the 
filament-plate  resistance,  are  given  by  equation  (21).  The  first 
mentioned  set  of  characteristics  is  shown  in  Fig.  74,  and  the 
second  set  in  Figs.  84  and  85.  It  is  important  to  note  that  in 
Fig.  74  each  characteristic  is  for  a  constant  plate-filament  voltage 
Ep,  while  in  Figs.  84  and  85  each  characteristic  is  for  a  constant 
plate  battery  voltage  Eb.  In  this  case,  as  was  explained  in  the 

1  Other  means  for  reducing  distortion  by  using  special  circuit  arrangements 
are  discussed  in  Section  78. 


THE  THERMIONIC  AMPLIFIER 


171 


previous  paragraph,  the  filament-plate  voltage  Ep  is  variable, 
having  a  different  value  for  each  adjustment  of  the  grid  voltage 
EffJ  due  to  the  varying  voltage  drop  in  the  external  resistance. 
It  is  obvious  that  if  the  external  plate  circuit  contains  reactance 


ItC 

0 


15  10 

Oriel  Volte 

FIG.  84. 


as  well  as  resistance,  the  static  characteristics  take  the  same  shape 
.as  when  the  reactance  is  zero,  being  determined  only  by  the 
resistance  component  of  the  external  impedance. 

Now,  the  thermionic  tube  is  used  mostly  in  a-c.  circuits.     It 


172 


THERMIONIC  VACUUM  TUBE 


is  therefore  necessary  to  know  the  shape  of  the  characteristics 
that  obtains  when  varying  potentials  are  impressed  on  the  grid, 
that  is,  it  is  necessary  to  know  the  shape  of  the  dynamic  character- 
istic. It  is  convenient  to  distinguish  three  cases:  (1)  the  dynamic 
characteristic  of  the  tube  itself;  (2)  that  of  the  plate  circuit 
containing  tube  and  non-inductive  resistance,  and  (3)  that  of 
the  circuit  containing  tube  and  impedance. 

(1)  The  first  can  be  determined  with  a  circuit  such  as  that 
shown  in  Fig.  72,  provided  that  the  resistance  TQ  is  zero  and  the 
current  meter  A  has  a  resistance  which  is  negligibly  small  com- 


FIG.  85. 


pared  with  the  plate  resistance.  As  far  as  the  passage  of  electrons 
from  cathode  to  anode  is  concerned,  the  thermionic  tube,  which 
operates  with  a  pure  electron  discharge,  shows  no  lag,  such  as  is 
found  to  exist  in  an  arc  which  depends  for  its  operation  on  ioniza- 
tion  by  collision  of  the  contained  gas  or  vapor.  The  only  reactance 
jK>ssessed  by  the  tube  is  capacitive  and  is  due  to  the  electrostatic 
capacity  between  the  electrodes.  It  is,  therefore,  in  effect,  a 
capacity  shunted  across  the  plate  resistance.  The  capacity  of 
v>rdinary  tubes,  is,  however,  so  small  (of  the  order  of  a  few  centi- 
meters) that  this  parallel  reactance  can  be  regarded  as  practically, 
infinite  for  frequencies  ranging  up  to  several  hundred  thousand 
cycles  per  second.  Hence,  for  this  range  of  frequencies  the 


THE  THERMIONIC  AMPLIFIER 


173 


dynamic  characteristic  of  the  tube  coincides  with  its  static  char- 
acteristic. 

(2)  If  the  external  resistance  r0  (Fig.  72),  instead  of  being 
zero,  has  a  finite  value  and  is  non-inductive,  the  dynamic  charac- 
teristic still  coincides  with  the  static  characteristic,  but  they  are 
different  from  the  characteristic  of  the  tube  itself,  being  given  by 
Figs.  84  and  85  instead  of  those  shown  in  Fig.  74. 

The  effect  of  the  external  non-inductive  resistance  on  the 
characteristic  of  the  output  circuit,  when  an  alternating  potential 
is  impressed  on  the  grid,  can  be  explained  as  follows:  Referring 


FIG.  86. 

to  Fig.  86,  let  the  three  parabolic  curves  represent  the  characteris- 
tics of  the  tube  itself,  the  middle  one  of  which,  let  us  say,  is  the  one 
obtained  when  the  plate-filament  voltage  has  a  definite  value  Ep. 
The  other  two  are  the  characteristics  for  higher  and  lower  values 
of  Ep.  Let  the  tube  be  inserted  in  the  circuit  shown  in  Fig.  72. 
Let  the  constant  grid  battery  voltage  Ett  be  so  adjusted  that  the 
direct  current  in  the  plate  circuit,  as  measured  with  A,  is  mo. 
Now,  on  account  of  the  voltage  drop  in  TO,  due  to  the  current  Ip 
in  it,  the  plate-filament  voltage  is  Ep  =  Eb—rolp.  If  Ip  be  varied 
by  impressing  an  alternating  potential  on  the  grid,  Ep  varies 
accordingly  since  E*  is  constant.  Thus,  if  the  negative  grid  poten- 
tial is  decreased  the  plate  current  increases.  This  causes  Ep  to 
decrease  to  the  value,  say,  corresponding  to  the  lower  characteris- 
tic shown  in  Fig.  86,  and  the  current  instead  of  increasing  to  a', 


174 


THERMIONIC  VACUUM  TUBE 


as  it  would  if  Ep  remained  constant,  increases  only  to  a.  For  the 
same  reason,  when  the  negative  grid  potential  is  increased  the  cur- 
rent decreases  only  to  b  instead  of  to  &'.  The  characteristic  there- 
fore straightens  out  and  takes  the  shape  given  by  boa,  instead  of 
b'oa'. 

Referring  to  equation  (20),  t  will  be  seen  hat  if  we  represent 
the  alternating  plate  voltage  and  current  by  ep  and  ip)  respectively, 
we  have  ep  =  —ipro.  The  plate  current  and  plate  voltage  are  there- 
fore 180°  out  of  phase.  The  plate  current  is,  however,  in  phase 
with  the  grid  potential,  so  that  the  grid  and  plate  potentials 
differ  in  phase  by  180°. 


FIG.  87. 

(3)  Let  the  plate  circuit  now  contain  reactance  as  well  as 
resistance,  that  is,  let  it  contain  an  impedance  Zo=ro+jxo.  Here 
we  have  ep=  —ipZ^  but  on  account  of  the  reactance  XQ  in  the  plate 
circuit  the  phase  difference  between  the  plate  and  grid  potentials 
may  differ  from  180°.  When  this  happens  the  dynamic  character- 
istic of  the  plate  circuit  takes  the  form  of  a  loop.  To  explain 
this  we  can  make  use  of  the  theorem  stated  on  page  157,  that  a 
voltage  eg  applied  between  filament  and  grid  is  equivalent  to  an 
electromotive  force  neg  impressed  on  the  plate  circuit,  where  M 
is  the  amplification  constant  of  the  tube.  The  phase  relations 

are  shown  in  Fig.  87  for  various  values  of  the  angle  0  =  tan~1— 
of  the  external  impedance.     Let  the  plate  current  be  represented 


THE  THERMIONIC  AMPLIFIER 


175 


by  ip  in  the  direction  OQP.  The  voltage  drop  iprp,  in  the  tube, 
due  to  its  plate  resistance,  is  given  by  OQ.  The  drop  ipZo  in  the 
external  impedance  ZQ  is  given  by  Qa.  Thus,  in  the  case  in  which 
the  angle  <j>  is  45°,  iPZQ=Qa2,  and  is  the  vector  sum  v*o  and  ipXo, 
the  total  driving  E.M.F.,  ^ea  in  the  plate  circuit  is  in  this  case 
given  by  Oa^.  Now  ep  is  equal  to  —ipZo  and  is  given  by  Oc2, 


Grid  Volfs 

FIG.  88. 

which  is  parallel  to  Qaz.  The  phase  difference  between  ev  and 
peg  or  eg  is  therefore  equal  to  the  angle  a^Oc^  which  is  157.5°. 
This  is  for  the  case  in  which  the  external  impedance  Z0  is  numeri- 
cally equal  to  the  plate  resistance  rP  (OQ  =  Qd2),  and  has  an  angle 
of  45°. 

Referring  now  to  Fig.  88,  let  the  negative  grid  battery  voltage 


176  THERMIONIC  VACUUM  TUBE 

be  equal  to  MS,  so  that  we  operate  around  the  point  0  of  the  tube 
characteristic  A  OB,  which  corresponds  to  the  plate  potential 
which  obtains  when  the  alternating  potential  impressed  on  the 
grid  is  zero.  Tne  other  two  tube  characteristics  are  for  the  maxi- 
mum and  minimum  potentials  which  the  plate  acquires  when  an 
alternating  potential  e  sin  pt  is  superimposed  on  the  constant 
negative  grid  potential  Eg  =  MS.  If  we  now  plot  the  plate  cur- 
rent as  a  function  of  the  varying  grid  potential  eff,  considering  at  the 
same  time  that  eg  and  the  alternating  plate  potential  ep  are  157.5° 
out  of  phase,  we  obtain  the  loop  shown  in  Fig.  88.  The  loop  is, 
of  course,  due  to  the  reactance  in  the  external  circuit,  because  there 
is  no  lag  within  the  tube.  This  loop  is  not  an  ellipse,  but  has  a 
curved  axis  CD,  the  general  slope  and  curvature  of  which  depends 
upon  the  angle  between  eg  and  ep,  which  in  turn  depends  upon  the 
angle  $  of  the  external  impedance.  As  <>  decreases  the  loop  nar- 
rows down,  its  axis  straightens  out  and  rotates  in  a  clock-wise 
direction  until,  when  <j>  is  zero,  that  is,  when  the  external  circuit 
contains  only  non-reactive  resistance,  the  loop  degenerates  into 
the  line  EF,  which  is  the  non-reactive  dynamic  characteristic  boa 
shown  in  Fig.  86.  It  will  be  observed  that  if  the  angle  6  between 
eg  and  ep  is  157.5°,  the  axis  of  the  loop  very  nearly  coincides  with 
the  approximately  straight  line  EF  obtained  when  6  is  180°. 

The  angle  6  depends  not  only  on  0,  the  angle  of  the  external 
impedance  ZQ,  but  also  on  the  value  of  this  impedance  compared 
with  the  plate  resistance  rP.  Thus  if  </>  is  90°  then  ipZo  is  given 
by  Qct4,  and  if  ZQ  is  numerically  equal  to  rp,  the  angle  a^Oc* 
between  ea  and  ep  is  0=  135°.  But  if  Z0=3rP,  the  angle  8  is  about 
160°.  In  this  case  also  the  axis  of  the  dynamic  characteristic 
coincides  very  nearly  with  the  line  EF  which  is  obtained  when 
0=180°. 

It  is  important  to  note  the  conditions  that  must  be  secured 
to  make  the  axis  of  the  dynamic  characteristic  approach  a 
straight  line.  While  the  curvature  of  the  characteristic  enables 
the  thermionic  tube  to  perform  certain  very  important  functions, 
such  as  detection  and  modulation  of  oscillating  currents,  it  is 
nevertheless  an  undesirable  feature  when  the  tube  operates  as 
an  amplifier.  It  follows  from  the  explanation  given  in  Section  57 
that  unless  the  characteristic  is  straight  the  output  current  wave  is 
not  an  exact  enlarged  reproduction  of  the  wave  impressed  on  the 
input.  This  causes  distortion  when  amplifying  telephonic  cur- 


THE  THERMIONIC  AMPLIFIER  177 

rents,  and  to  avoid  it  the  amplifier  must  be  operated  under  such 
conditions  that  its  characteristic  is  substantially  linear.  Now, 
it  will  be  shown  later  that  when  operating  the  tube  as  an  ampli- 
fier, maximum  power  amplification  is  obtained  when  the  external 
impedance  is  numerically  equal  to  the  plate  resistance  of  the  tube. 
If  this  equality  is  preserved  and  the  angle  of  the  external  imped- 
ance is  not  greater  than  about  45°,  the  axis  of  the  characteristic 
is,  as  we  have  seen,  substantially  linear  over  a  considerable  range 
of  input  voltage.  In  practice  the  conditions  are  often  even  better, 
because  the  angle  of  the  external  impedance  is  often  much  less  than 
45°.  This  is,  for  example,  the  case  where  the  tube  is  operated  as 
a  telephone  repeater;  the  secondary  of  the  output  transformer 
feeds  into  a  long  line  of  comparatively  high  resistance,  so  that  the 
angle  of  the  effective  impedance  into  which  the  tube  works  is  very 
small. 

In  cases  where  the  angle  of  the  external  impedance  is  neces- 
sarily large,  we  can  still  secure  a  practically  linear  axis  for  the 
dynamic  characteristic  by  making  the  external  impedance  larger 
than  the  plate  resistance.  We  would  therefore  gain  in  quality  of 
transmission  at  the  expense  of  amplification.  But  the  necessary 
sacrifice  in  amplification  would  not  be  large.  Although  maximum 
amplification  is  secured  when  the  external  impedance  is  equal  to 
the  plate  resistance,  the  decrease  in  amplification  is  small  even 
when  the  external  impedance  is  twice  as  large  as  the  plate  resist- 
ance (see  Fig.  112). 

If  the  necessary  precautions  be  taken  to  secure  the  conditions 
necessary  to  make  the  axis  of  the  dynamic  characteristic  substan- 
tially linear,  we  can  extend  the  theorem  deduced  on  page  157 
from  the  stray  field  relation:  A  voltage  eg  applied  between 
filament  and  grid  establishes  a  current  in  the  plate  circuit  which 
is  given  by 

......  '  (22) 


where  ju  is  the  amplification  constant  of  the  tube  rp  its  plate  resist- 
ance and  ZQ  the  external  impedance  in  the  plate  circuit. 

If  the  conditions  are  not  such  as  to  make  the  characteristic 
linear  this  equation  is  still  true  as  far  as  the  fundamental  frequency 
is  concerned,  but  the  curvature  of  the  characteristic  introduces 
harmonics  which  would  necessitate  the  addition  of  terms  of  higher 
order  of  smallness  to  equation  (22). 


17$  THERMIONIC  VACUUM  TUBE 

The  theorem  embodied  in  eauation  (22)  is  of  fundamental 
importance  and  is  instrumental  in  the  solution  of  many  vacuum 
tube  problems.  We  shall  have  occasion  to  make  extensive  use 
of  it  in  what  follows.  t 

60.  Conditions  for  Distortionless  Amplification.  Distortion- 
less amplification  is  obtained  if  the  amplified  current  in  the  output 
circuit  is,  for  the  whole  range  of  frequencies  which  it  is  desired  to 
transmit,  an  exact  enlarged  reproduction  of  the  input  current. 
Distortion  can  be  produced  in  two  ways:  (1)  When  currents  of 
different  frequencies  are  not  amplified  in  the  same  proportion; 
(2)  when  the  amplification  is  not  independent  of  the  input  voltage. 

(1)  As  far  as  the  first  effect  alone  is  concerned,  the  amplifica- 
tion will  be  distortionless  if  the  whole  circuit  is  non-reactive.     The 
circuits  commonly  used  in  connection  with  the  tube  are  not  non- 
reactive,  but  the  necessary  transformers  and  condensers  can  always 
be  so  chosen  that  for  the  operating  range  of  frequencies  the  total 
impedance  is  not  unduly  affected  by  the  frequency.     As  far  as 
the  tube  itself  is  concerned  it  is  to  be  noted  that  the  capacities 
between  the  electrodes  introduces  a  reactance  effect.     Of  these  we 
distinguish  between  the  capacity  between  filament  and  plate,  and 
the  effective  input  impedance  as  measured  between  the  filament 
and  grid.     When  the  amplification  is  expressed  in  terms  of  the 
potential  actually  applied  to  the  grid,  the  only  inter-electrode 
capacity  that  comes  into  consideration  is  the  capacity  between 
filament  and  plate.    This  is  so  small  that  when  the  amplification 
is  expressed  in  this  way  it  is  found  to  be  independent  of  the  fre- 
quency for  frequencies  ranging  up  to  several  hundred  thousand 
cycles  per  second.     The  power  amplification  is  usually  expressed 
in  terms  of  the  ratio  of  the  power  developed  in  the  external  output 
circuit  to  the  total  power  impressed  on  the  input.     In  this  case 
the  effective  reactance  due  to  the  inter-electrode  capacities  depends 
on  the  circuit  used,  as  will  be  explained  in  Sections  69  to  71.    Under 
the  conditions  under  which  amplifiers  are  mostly  operated,  the 
electrode  capacities  usually  have  a  very  small  effect.     The  general 
effect,  however,  is  to  decrease  the  amplification  when  the  frequency 
becomes  very  high. 

(2)  The  second  condition  for  distortionless  amplification  will 
not  be  satisfied  unless  the  axis  of  the  dynamic  characteristic  of 
the  output  circuit  is  linear  over  the  operating  range  of  voltage. 
As  was  shown  in  the  previous  Section,  this  can  be  secured  by 


THE  THERMIONIC  AMPLIFIER  179 

making  the  external  impedance  in  the  output  circuit  sufficiently 
large. 

It  is  important  to  note  that  another  condition  for  distortionless 
amplification  is  that  the  input  voltage  must  be  kept  within  certain 
limits  determined  by  the  d-c.  plate  and  grid  voltages  and  the  struc- 
ture of  the  tube.  Let  us  assume  that  the  external  impedance  is 
sufficiently  large  to  straighten  out  effectively  the  characteristic. 
The  question  now  is  what  range  of  input  voltage  can  be  employed 
without  overtaxing  the  tube.  If  the  input  voltage  is  so  large  that 
the  grid  becomes  sufficiently  positive  to  take  appreciable  current, 
the  positive  halves  of  the  output  wave  can  be  reduced  in  the  man- 
ner explained  in  Section  57.  This  reduction  is  more  marked  the 
larger  the  external  impedance  in  the  output  circuit,  because  the 
extent  to  which  the  grid  can  become  positive  without  taking  appre- 
ciable current  depends  on  the  potential  difference  Ep  existing 
between  filament  and  plate  at  the  moment  that  the  grid  is  positive 
and  on  the  structure  of  the  tube.  Remembering  that  the  stray 
field  between  filament  and  grid,  due  to  the  potential  difference 
Ep,  tends  to  draw  the  electrons  through  the  openings  of  the  grid, 
it  will  be  seen  that  the  larger  Ep  the  higher  must  be  the  positive 
grid  voltage  to  overcome  the  stray  field  and  attract  the  electrons 
to  the  grid.  Now  the  external  impedance  has  the  effect  of  decreas- 
ing the  plate-filament  potential  difference  when  the  flow  of  electrons 
from  plate  to  filament  through  the  impedance  is  increased,  that  is, 
during  the  half  .cycle  when  the  grid  is  positive.  This  reduces  the 
stray  field  and  consequently  increases  the  flow  of  electrons  to  the 
grid.  This  is  the  effect  that  gives  rise  to  the  bend  C  in  the  dynamic 
characteristic  shown  in  Fig.  86.  If  we  say  that  g  is  the  positive 
potential  with  respect  to  the  filament  which  the  grid  can  acquire 
without  taking  appreciable  current,  we  can  state  that  one  condition 
for  distortionless  amplification  is 

e<-   |  Eg+e  |  +   \g  | (23) 

4* 

where  Eg  is  the  voltage  of  the  grid  battery,  e  the  peak  value 
of  the  input  voltage,  and  e  the  intrinsic  potential  difference 
between  filament  and  grid. 

Another  condition  is  that  the  peak  value  of  the  input  voltage 
must  not  exceed  the  value  given  by  mn  (Fig.  86),  otherwise  the 
negative  peaks  of  the  output  current  wave  will  be  chopped  off. 


180  THERMIONIC  VACUUM  TUBE 


Now  sn  is  given  by  —  -,  where  E'p  is  the  potential  difference  between 

filament  and  plate  at  the  moment  when  the  grid  has  its  maximum 
negative  value,  and  sm  is  the  voltage  Ea  of  the  grid  battery. 
We  therefore  have  the  two  conditions: 


I   Eg+e 

— P+      I 
M 


(24) 


or  when  the  tube  is  working  at  full  capacity,  that  is,  when  operating 
over  the  whole  range  of  the  characteristic. 


e=-   I  Eg+e  I  +   \g  I    = 


(25) 


61.  Amplification  Equations  of  the  Thermionic  Amplifier.  We 
shall  now  derive  quantitative  expressions  for  the  amplification 
produced  by  the  three-electrode  thermionic  tube.  It  will  be 
recognized  that  when  operating  as  a  power  amplifier  the  tube 
derives  the  extra  power  from  the  d-c.  battery  inserted  in  the 
plate  circuit.  The  energy  of  the  plate  battery  is  released  by 
the  influence  of  the  grid  potential  on  the  current  in  the  plate 
circuit  and  the  amount  of  power  released  depends  almost  entirely 
on  the  influence  of  the  grid  potential.  , 

In  deriving  the  following  equations  we  assume  that  the  grid 
is  maintained  sufficiently  negative  with  respect  to  the  filament  to 
prevent  any  appreciable  current  convection  between  filament 
and  grid;  that  is,  the  tube  will  be  assumed  to  operate  within  the 
limits  defined  by  equations  (24)  and  (25).  We  shall  also  assume 
that  the  impedance  conditions  in  the  plate  circuit  are  such  as  to 
make  the  characteristic  of  the  plate  circuit  substantially  linear 
over  the  operating  range  of  voltages.  These  conditions  can  very 
nearly  be  satisfied  in  practice  even  when  the  circuit  constants 
are  so  adjusted  as  to  give  a  maximum  degree  of  amplification. 
Under  these  conditions  the  alternating  current  iv  in  the  plate 
circuit  is  related  to  the  alternating  potential  ea,  applied  to  the 
grid,  by  equation 


(22) 


THE  THERMIONIC  AMPLIFIER 


181 


where  rp  is  the  plate  resistance  and  ZQ  the  external  impedance. 
This  equation  enables  us  to  derive  the  amplification  equations  in 
a  very  simple  manner. 

62.  Voltage  Amplification.  Consider  first  the  case  in  which 
the  tube  is  used  as  a  voltage  amplifier.  The  voltage  developed  in 
the  impedance  ZQ  is  eo=ipZo,  which  according  to  equation  (22) 
becomes: 


rp+Z0' 
and  the  voltage  amplification  is  therefore 


=== 


rp+Z0' 


(26) 


It  must  be  noted  that  eg  is  the  a-c.  potential  difference  actually 
established  between  filament  and  grid. 

It  will  be  seen  that  //  increases  as  ZQ  is  increased  and  asymptot- 
ically approaches  the  maxi- 
mum value  /*  when  ZQ  becomes 
infinitely  large  compared  with 
rp.  The  constant  which  de- 
pends on  the  structure  of 
the  tube  and  determines  the 
stray  field,  is  therefore  simply 
the  maximum  voltage  ampli- 
fication obtainable  from  the 
tube.  When  a  tube  is  to  be 
used  as  a  voltage  amplifier  it 
should  therefore  be  designed 
to  have  as  high  a  value  of  /* 
as  possible.  Fig.  89  shows  a 
Western  Electric  voltage  am- 
plifier. The  amplification  con- 
stant ju  of  this  tube  is  40. 

A  voltage  amplification  of 
several  hundred  fold  is  not 
hard  to  obtain,  it  being  simply 
necessary  to  design  the  tube 
accordingly,  since  ju  is  a  struc- 
tural constant.  In  using  tubes 
however,  necessary  to  consider 


FIG.  89. 

as   voltage   amplifiers    it   is, 
also    the    other   factors    that 


182 


THERMIONIC  VACUUM  TUBE 


influence  the  voltage  amplification.  For  example,  it  follows 
directly  from  equation  (26)  that  the  external  impedance  should 
be  made  several  times  as  large  as  the  plate  resistance  of  the  tube. 
Now,  for  the  same  amount  of  filament  surface  the  plate  resistance 
increases  approximately  as  the  square  of  /z  (see  equation  15) 
and  may  acquire  such  a  high  value  as  to  necessitate  an  impracti- 
cably high  external  impedance.  It  is,  therefore,  often  necessary, 
when  increasing  //,  to  increase  the  amount  of  filament  surface  so 
as  to  reduce  the  plate  resistance  as  much  as  possible.  It  is,  of 
course,  also  possible  to  decrease  the  plate  resistance  by  increasing 
the  d-c.  plate  voltage,  provided  we  do  not  operate  beyond  the 
minimum  saturation  voltage. 

Referring  now  to  equation  (26)  let  ZQ=TQ-}-JXQ',    the  voltage 
amplification  is  then  given  by 


(27) 


Suppose  the  tube  is  inserted  in  the  circuit  shown  in  Fig.  90, 
and  that  it  is  desired  to  obtain  the  voltage  developed  between 


FIG.  90. 


the  ends  A  and  B  of  the  impedance  ZQ.  This  voltage  CQ  can  be 
measured  by  connecting  an  electrostatic  voltmeter  between  A 
and  B.1  The  secondary  of  the  transformer  T  can  be  wound  to 
have  as  high  an  impedance  as  possible,  thus  impressing  the  highest 
possible  voltage  eg  on  the  grid  for  a  given  voltage  in  the  primary 
ofT. 

Let  us  now  consider  the  two  extreme  cases  in  which  ZQ  is  (1) 
a  non-inductive  resistance  TO  (zo=0)  and  (2)  a  practically  pure 

1  A  thermionic  tube  can  be  used  as  an  electrostatic  voltmeter  in  the  manner 
shown  in  Section  114. 


THE  THERMIONIC  AMPLIFIER 


183 


reactance  XQ  (ro=0).     In  the  first  case  the  voltage  amplification 
is  given  by 


(28) 


The  relation  between  —  and  —  is  shown  by  curve  II  of  Fig.  91 

Cg  Tp 

from  which  it  is  seen  that  —  reacnes  about  90  per  cent  of  its 

e, 

maximum  value  M  when  r0=  10  rp.     (In  computing  these  curves  u 
was  taken  equal  to  10.) 


I 

"5 

o 

t6 

CO 


n 


h 

rp 

FIG.  91. 


If  ZQ  is  a  pure  reactance  XQ,  the  voltage  amplification  is  given  by 


eo_ 


(29) 


Curve  I  of  Fig  91  shows  the  relation  between  —  and  — .      It  is 

eg          rp 

seen  that  there  is  a  distinct  advantage  in  making  the  tube  work 
into  a  reactance,  the  voltage  amplification  rising  to  about  90  per 
cent  of  its  maximum  value  when  XQ  is  numerically  only  twice  rp. 
It  is,  however,  advisable  to  make  the  reactance  as  large  as  possible 
in  order  to  minimize  distortion  due  to  the  curvature  of  the  charac- 
teristic. The  use  of  a  reactance  instead  of  a  resistance  has  another 


184  THERMIONIC  VACUUM  TUBE 

advantage.  If  Zo  is  a  pure  resistance  and  several  times  greater 
than  the  plate  resistance,  a  considerable  portion  of  the  voltage 
of  the  plate  battery  is  lost  in  ro,  so  that  to  secure  the  necessary 
potential  difference  between  filament  and  plate  it  would  be  neces- 
sary to  use  a  rather  high  plate  battery  voltage.  This  can  be 
avoided  by  using  instead  of  a  pure  resistance  a  choke  coil  which 
has  a  comparatively  small  d-c.  resistance. 

On  the  other  hand,  the  value  of  the  tube  as  a  voltage  amplifier 
lies  in  the  fact  that  it  can  be  operated  in  a  non-inductive  circuit, 
and  in  this  respect  it  performs  an  important  function,  in  that  it 
serves  the  purpose  of  producing  high  degrees  of  amplification 
with  very  little  distortion.  Unlike  the  transformer,  for  example, 
it  furnishes  a  voltage-amplifying  means  that  is  independent  of 
frequency  unless  the  frequency  is  very  high.  And,  as  a  matter  of 
fact,  it  can  also  be  used  to  produce  power  amplification  that  is 
practically  independent  of  frequency. 

When  several  tubes  are  used  in  cascade  formation  in  multi- 
stage non-inductive  amplifier  sets,  all  but  the  last  tube  should  be 
used  as  voltage  amplifiers,  because  the  tube  is  a  potential  operating 
device.  It  works  best  as  an  amplifier  when  its  grid  does  not  take 
appreciable  current;  that  is,  when  the  tube  operates  within  the 
limits  defined  by  equations  (24)  and  (25).  The  input  power 
consumed  by  the  tube  is  therefore  usually  very  small,  and  all 
that  is  necessary  is  to  make  the  input  voltage  applied  between 
filament  and  grid  as  high  as  possible. 

It  must  be  pointed  out  that  unless  it  is  necessary  to  use  a 
non-inductive  circuit  it  is  best  to  operate  all  tubes  in  a  multi- 
stage amplifier  set  as  power  amplifiers,  and  use  voltage  step-up 
transformers  between  the  tubes.  Consider,  for  example,  the 
circuit  in  Fig.  92.  (The  circuits  shown  here  do  not  include 
details  that  are  necessary  to  give  best  operation  in  practice. 
They  are  merely  skeleton  circuits  intended  to  illustrate  the 
points  under  consideration.  Complete  circuits  will  be  discussed 
below.) 

If  the  tube  A  were  to  be  used  as  a  voltage  amplifier,  it  would 
be  necessary  to  make  ZQ  several  times  as  large  as  rp.  This  does 
not,  however,  give  maximum  total  amplification,  because  when 
using  transformers  we  have  to  consider  the  power,  and  maximum 
amplification  is  obtained  when  ZQ=rp,  T2  being  used  as  a  voltage 
step-up  transformer.  This  can  be  shown  as  follows:  The  power 


THE  THERMIONIC  AMPLIFIER 


185 


in  ZQ  will  be  a  maximum  for  maximum  voltage  e'g  impressed 
on  the  input  of  the  second  tube  B.  Now,  the  voltage  eo  in  ZQ  is 
given  by: 


where  ju  is  the  amplification  constant  of  tube  A.  Now,  the  voltage 
ratio  of  the  transformer  T%  is  -y^  •  Hence,  the  voltage  impressed 
on  tube  B  is: 

e'<=  (rp+ZQ)° (30) 


FIG.  92. 

It  is  in  all  cases  desirable  to  make  Z\  as  large  as  can  possibly 
be  done  in  practice.  Hence,  regarding  Z\  as  fixed  and  differen- 
tiating e'g  with  respect  to  ZQ  and  equating  to  zero,  it  will  be 
seen  that  e'g  is  a  maximum  when  Zo=rp,  and  this,  it  will  be  shown 
in  the  next  paragraph,  is  the  condition  for  maximum  power  in  ZQ. 

63.  Power  Amplification.  The  three-electrode  thermionic  tube 
can  be  used  to  amplify  power,  and  in  this  property  lies  its  great 
usefulness.  It  is  its  amplifying  property  that  enables  it  to  be 
used  also  as  an  oscillation  generator.  There  are  other  types  of 
amplifiers,  such  as,  for  example,  the  arc  which  amplifies  in  virtue 
of  its  negative  resistance  characteristic  and  therefore  operates  on 
an  entirely  different  principle.  But  the  thermionic  amplifier,  or 
audion,  has  certain  marked  advantages  over  other  types.  Unlike 
the  arc  it  does  not  depend  for  its  operation  on  ionization  by  col- 
lision of  residual  gas,  and  in  fact  operates  satisfactorily  only  when 
the  vacuum  is  so  high  that  ionization  by  collision  plays  a  negligibly 
small  part  in  current  convection  in  the  tube.  The  discharge  is 
therefore  steady  and  reproducible.  When  using  the  device  as  a 


186  THERMIONIC  VACUUM  TUBE 

telephone  relay,  for  example,  steadiness  and  reproducibility  are 
conditions  that  must  be  complied  with,  and  this  is  also  true  of  many 
other  cases  where  amplifiers  are  used.  It  is  furthermore  capable 
of  amplifying  currents  of  frequencies  ranging  all  the  way  up  to 
several  million  cycles  per  second,  and  if  properly  designed  it  can 
be  made  to  produce  an  extraordinarily  high  degree  of  amplifica- 
tion. I  have  for  example,  obtained  with  a  specially  designed  tube, 
a  power  amplification  of  3000-fold. 

An  equation  for  the  power  amplification  can  be  obtained 
directly  from  the  equations  deduced  above.  Let  us  consider 
the  circuit  shown  in  Fig.  90.  It  is  desired  to  amplify  the  power 
in  the  transformer  T  which  may  be  at  the  end  of  a  section  of  tele- 
phone line  or  may,  for  example,  be  connected  in  the  output  of 
the  generator  G.  Let  the  a-c.  potential  impressed  on  the  grid 
be  eff.  Then  the  alternating  current  iv  in  the  output  circuit 
FPABis 


v 

where  rp  is  the  plate  resistance.    The  voltage  CQ  in  ZQ  is 
and  hence  the  power  in  ZQ  is 

_A*VZ0COS 


where  cos  <£  is  the  power  factor. 

In  order  to  get  the  power  amplification  it  is  necessary  also  to 
know  the  power  expended  in  the  input.  The  grid  current  does  not 
bear  a  simple  relation  to  the  operating  parameters,  but  to  get  an 
indication  of  how  current  in  the  grid  circuit  affects  the  amplifica- 
tion, we  can  expand  the  obvious  functional  relationship,  I0=f(Ep, 
Eg)  into  a  Taylor  series,  thus: 


}-8lg=f(Ep+dEp, 


THE  THERMIONIC  AMPLIFIER  187 

second  and  higher  order  quantities  being  neglected.     By  making 
the  following  substitutions: 


8Eg=eg, 
we  get: 


Putting  —  =  /*i,  the  input  power  becomes,  if  we  neglect  the  power 

6g 

consumption  in  the  input  transformer: 


Hence  the  power  amplification  becomes: 


=  e0p  cos  <>  =         M0  cos 
effiff         ( 


This  equation  shows  how  the  power  amplification  is  affected 
by  the  grid  conductance  g0,  the  amplification  factor  p  and  the 
reflex  mutual  conductance  gn.  For  a  perfectly  unilateral  amplifier 
the  output  circuit  has  no  effect  on  the  input  and  then  gn  =  0. 

Conditions  can  readily  be  realized  in  practice  which  make 
both  gn  and  gg  negligibly  small.2  Conditions  under  which  they 
become  appreciable  will  be  discussed  in  Section  69.  If  we  neglect 
these  quantities  the  input  resistance  is  infinite  and  the  power  loss 
in  the  input  indeterminate.  We  can,  however,  shunt  the  input 

1  This  equation  is  equivalent  to  that  derived  by  Latour.     (Electrician, 
Dec.,  1916.) 

2  When  there  are  reactive  effects,  as,  for  example,  when  the  output  circuit 
is  reactive,  we  should,  strictly  speaking,  consider  the  mutual  admittance 
and  reflex  mutual  admittance  instead  of  simply  the  mutual  conductances, 
because  under  these  conditions  the  grid  potential  and  plate  current  are  out 
of  phase.     The  mutual  admittances  are  then  complex  quantities  involving 
the  mutual  conductances  and  the  mutual  susceptances.     (See  Fig.  87.)     When 
the  circuit  constants  are  so  proportioned  that  the  axis  of  the  dynamic  char- 
acteristic is  substantially  linear,  which  is  the  condition  for  distortionless 
transmission,  the  angle  of  the  mutual  admittance  is  so  small  that  we  can, 
to  a  first  approximation,  neglect  the  mutual  susceptances. 


188  THERMIONIC  VACUUM  TUBE 

with  a  resistance  ra  (Fig.  90),  as  was  suggested  by  H.  D.  Arnold, 
and  so  proportion  its  value  that  the  input  transformer  works  most 
efficiently.  The  power  expended  in  this  resistance  can  then  be 
taken  as  a  measure  of  the  input  power.  In  telephone  repeater 
circuits  this  resistance  usually  has  a  value  of  about  600,000  ohms. 
Equation  (32)  then  becomes: 


cos 


or  putting  Zo=ro+jx0,  the  power  amplification  can  be  expressed 
as: 


(33a) 

Let  us  first  consider  the  case  in  which  ZQ  takes  the  form  of  a  non- 
inductive  resistance  (zo=0).  The  power  amplification  is  then 
simply  given  by 


(34) 


and  it  will  be  seen  by  differentiating  rj  with  respect  to  ro  and 
equating  the  derivative  to  zero,  that  the  power  amplification  is  a 
maximum  when  ro=rP. 

For  the  general  case  in  which  the  reactance  XQ  is  not  zero,  we 

note  that  0  =  tan-1—  and  cos  <ft=    .     °  Substituting  this 


in  equation  (33)  the  power  amplification  becomes: 

XORX 


This  is  also  a  maxunum  when  Z0  is  numerically  equal  to  rp,  as  is 

shown  in  Fig.  112,  page  220,  where  the  curve  represents  the  relation 

g 

between  the  power  amplification  and  the  ratio  —  for  the  case 

rp 

in  which  angle  of  the  external  impedance  Z0  is  45  °,  that  is, 

tan  <£  =  -9=  l.     When  the  tube  is  used  for  the  purpose  of  amplify- 

7*0 

ing  telephonic  currents,  the  energy  is  translated  into  sound  waves 
through  the  motion  of  the  receiver  diaphragm.     In  such  cases, 


THE  THERMIONIC  AMPLIFIER 


189 


•f 

it  will  be  shown  later,  the  ratio  —  can  deviate  considerably  from 

unity  before  resulting  in  any  serious  diminution  of  the  effect 
produced  upon  the  organs  of  hearing. 

If  we  put  Z0=nrp  equation  (33)  becomes: 


_v^ra  cos  0 


(36) 


which  clearly  shows  the  importance  of  the  mutual  conductance 

-  or  the  steepness  of  the  plate  current-grid  voltage  characteristic. 
rv 

It  will  be  seen  later  that  'this  quantity  plays  an  equally  important 
role  in  the  operation  of  the  tube  as  oscillation  generator  and  radio 
detector. 


FIG.  93. 

64.  Experimental    Verification    of    Amplification    Equations. 

The  amplification  was  determined  experimentally  as  a  function 
of  the  tube  parameters  with  the  circuit  arrangement  shown  in 
Fig.  93.1  This  circuit  was  made  non-inductive  throughout. 
The  input  voltage  could  be  varied  by  means  of  the  resistance  r\ 
and  measured  with  a  Duddell  thermo-galvanometer  G\  and  resist- 
ance T2.  The  grid  battery  Eg  was  inserted  to  insure  that  the  tube 
was  always  operated  within  the  limits  given  by  equations  (24). 
Now  the  amplifier  is  always  operated  with  a  battery  in  the  plate 
circuit,  so  that  there  is  a  constant  direct  current  in  this  circuit 
whether  the  a-c.  input  be  applied  or  not.  The  application  of  the 
1 H.  J.  VAN  DER  BIJL,  Phys.  Rev.,  Vol.  12,  p.  194,  1918. 


190  THERMIONIC  VACUUM  TUBE 

a-c.  input  voltage  establishes  an  alternating  current  in  the  plate 
circuit  which  is  superimposed  upon  the  constant  direct  current. 
This  a-c.  could  not  be  measured  accurately  by  simply  inserting 
an  a-c.  meter  in  the  plate  circuit,  because  it  was  often  small  com- 
pared with  the  direct  current  that  would  constantly  flow  through 
the  galvanometer.  A  galvanometer  that  would  be  capable  of 
carrying  the  direct  current  would,  therefore,  not  be  sensitive  enough 
to  measure  the  increase  in  current  due  to  the  a-c.  input  with  any 
degree  of  accuracy.  On  the  other  hand,  it  was  not  possible  to 
separate  the  a-c.  from  the  d-c.  in  the  usual  way  with  appropriate 
inductances  and  capacities,  because  then  the  amplification  would 
be  influenced  in  a  large  measure  by  the  constants  of  the  circuit. 
For  these  reasons  the  balancing  scheme  was  used.  The  direct 
current  was  measured  with  the  milliammeter  £3  and  the  alternating 
current  with  the  thermocouple  and  milliammeter  Gz.  In  series 
with  (/2  was  a  battery  Bz  so  poled  that  when  the  input  voltage  was 
not  impressed  there  was  no  current  in  (72,  the  direct  current 
being  by-passed  through  Rl.  The  resistance  of  G<z  was  small 
compared  with  R1  so  that  practically  all  the  alternating  current 
established  in  the  plate  circuit  flowed  through  G?2.  It  is  evident 
that  the  effective  external  resistance  is  TQ.  The  whole  system  was 
carefully  shielded  and  care  was  taken  to  avoid  any  disturbing 
effects  due  to  mutual  and  shunt  capacity  of  the  leads  and  resist- 
ances. Such  precautions  were  necessary  because  the  frequency 
at  which  measurements  were  made  ranged  from  200  to  350,000 
cycles  per  second.  The  resistances  r\  and  rz  consisted,  for  exam- 
ple, of  thin  straight  wires  stretched  on  a  board. 

The  amplification  was  found  to  be  practically  independent  of 
frequency  over  the  range  mentioned  above.  The  input  voltage 
was  varied  from  a  few  hundredths  of  a  volt  to  several  volts.  Fig. 
94  shows  the  relation  between  the  voltage  in  ro  (the  output  voltage) 
and  voltage  as  measured  with  G\  and  r2  (the  input  voltage). 
In  these  measurements  ro  was  made  equal  to  the  plate  resistance 
of  the  tube.  The  linear  relation  obtained  shows  that  the  amplifica- 
tion is  independent  of  the  input  voltage,  a  result  which  justifies 
the  use  of  equation  (22). 

Equation  (28)  was  verified  by  measuring  the  output  voltage 
for  a  constant  input  voltage  and  different  external  resistance  TO. 
The  results  are  shown  in  Fig.  95  where  the  circles  indicate  the 
observed  values  and  the  curve  was  computed  from  equation  (28). 


THE  THERMIONIC  AMPLIFIER 


191 


The  abscissae  give  the  ratio  —  and  the  ordinates  the  voltage  ampli- 
fy 

fication  — .     The  value  of  p  for  this  tube  was  10.2,  and  the  input 
ea 

voltage  in  this  particular  experiment  was  3.55  volts.     In  experi- 
ments like  this  it  must  be  remembered  that  the  plate  resistance 


1,5 


3.0 
A.C.lnputjVolts 

FIG.  94. 


4.5 


6.0 


rp  of  the  tube  depends  upon  the  potential  difference  between  fila- 
ment and  plate  which,  if  the  voltage  of  the  plate  battery  remains 
constant,  changes  every  time  TQ  is  given  a  different  value.  In 
order  to  operate  the  amplifier  under  the  same  conditions  through- 
out the  experiment,  the  plate  battery  voltage  should  always  be 
adjusted  to  keep  the  plate  resistance  constant. 


192 


THERMIONIC  VACUUM  TUBE 


The  power  developed  in  the  external  resistance  TO  as  a  function 
of  r0  is  shown  in  Fig.  96.     The  plate  resistance  of  the  tube  was  kept 


constant  at  14,800  ohms.    The  power  in  ro  is  seen  to  be  a  maximum 
when  ro=  15,000  ohms,  which  is  in  close  argeement  with  equation 


Z4*IO' 


30  4< 

rQ(Ohms) 

FIG.  96. 


50 


60 


TOxlO3 


(34)  which  requires  maximum  power  amplification  when  ro=rv. 
The  input  in  this  experiment  remained  constant  and  corresponded 


THE  THERMIONIC  AMPLIFIER  193 

to  an  input  voltage  3.55  volts.  Putting  the  resistance  r<i  and 
that  of  the  galvanometer  G\  in  series  equal  to  rg  we  may  express 

e  2 
the  input  power  as  -— .     Combining  this  with  equation  (34)  the 

TQ 

power  developed  ro  is  given  by 

P  =  ff\v  (37) 

(r0+rp)2 

Now  fy  =  3.55  volts  and  ju  =  10.2,  and  if  we  put  rp  =  ro=15,000 
ohms  we  find  that  the  power  in  ro  according  to  the  above  equation 
is  22X10~3  watt,  which  is  in  good  agreement  with  the  observed 
value,  namely  23X10"3  watt. 

65.  Methods  of  Measuring  the  Amplification  Constant.  The 
amplification  constant  ju  can  be  measured  with  considerable 
accuracy  in  a  number  of  different  ways.  It  is  perhaps  the  most 
easily  determined  constant  of  the  tube. 

The  first  method  we  describe  is  not  the  most  accurate  nor  the 
simplest,  but  furnishes  a  clear  demonstration  of  the  significance  of 
M.  If  the  equation  of  the  characteristic  (equation  3)  be  differen- 
tiated partially  first  with  respect  to  the  plate  voltage  Ev  and  then 
with  respect  to  the  grid  voltage  Eg  it  will  be  found  that 

87 


n  can  therefore  be  obtained  by  measuring  the  slopes  of  the  Ip, 
Ep  and  Ip,  Eg  characteristics,  the  slopes  being  taken  at  corre- 
sponding points  of  the  two  curves.  In  obtaining  these  curves 
care  must  be  taken  that  the  plate  circuit  does  not  contain  an 
appreciable  resistance  which  would  influence  the  slopes  of  the 
characteristics. 

Instead  of  going  through  the  rather  tedious  process  of  taking 
the  characteristic  curves  and  then  measuring  their  slopes  at  the 
desired  points,  we  can  determine  /z  by  a  method  indicated  by 
equation  (8),  which  gives  a  linear  relation  between  the  plate 
and  grid  voltages  necessary  to  keep  the  plate  current  constant. 
As  Fig.  76  (p.  157)  indicates,  this  method  is  quite  accurate.  It  is 
of  course  not  necessary  to  obtain  more  than  two  corresponding 
values  of  plate  and  grid  voltages.  For  example,  let  the  plate 


194 


THERMIONIC  VACUUM  TUBE 


current  for  two  definite  values  of  EP  and  Eff  be  Ip.  Now  let  the 
plate  voltage  be  increased  to  E'p.  This  causes  an  increase  in  the 
plate  current.  In  order  to  bring  it  back  to  its  original  value  Ip> 
the  grid  must  be  made  more  negative  with  respect  to  the  filament. 
If  the  necessary  grid  voltage  is  E'g,  M  is  given  by 


(39) 


A  convenient  and  rapid  means  of  measuring  /i  is  shown  in  Fig. 
97.1  E\  is  a  battery  of  small  dry  cells  of  about  10  or  20  volts. 
By  closing  the  key  K  opposite  potentials  are  applied  to  the  grid  and 


FIG.  97. 

plate,  their  values  depending  upon  those  of  r\  and  r%.  Since  a 
potential  applied  to  the  grid  produces  ju-times  the  effect  of  a 
potential  applied  to  the  plate,  it  is  evident  that  no  change  will 
be  produced  in  the  reading  of  the  current  meter  by  closing  K  if 

—  =M-     For  convenience  in  measurement  r-z  is  given  a  fixed  value 

7*2 

of  10  ohms  and  n  consists  of  three  dial  rheostats  of  1000,  100 
and  10  ohms  arranged  in  steps  of  100,  10  and  1  ohms  each.  The 
rheostats  are  marked  in  tenths  of  the  actual  resistances,  so  that  the 
setting  of  the  dials  gives  n  directly. 

A  similar  method  has  also  been  described  by  J.  M.  Miller,2 
who  use,d  a  source  of  alternating  current  instead  of  the  battery 

1  H.  J.  VAN  DER  BIJL,  Proc.  I.R.E.,  Vol.  77,  p.  112,  1919. 
3  J.  M.  MILLER,  Free.  I.R.E.,  Vol.  6,  p.  141,  1918. 


THE  THERMIONIC  AMPLIFIER  195 

Ei.  The  meter  is  replaced  by  a  telephone  receiver  and  the  resist- 
ances n  and  r2  are  adjusted  until  the  tone  in  the  receiver  is  a  mini- 
mum. The  use  of  an  alternating  current  has  the  advantage  that 
it  also  allows  a  simple  determination  of  the  plate  resistance  of  the 
tube. 

66.  Measurement  of  the  Plate  Resistance.  We  have  seen 
that  the  characteristic  of  the  amplifier  can  to  a  first  approximation 
be  given  by  equation  (3),  in  which  the  exponent  is  2.  For  such  a 
characteristic  the  plate  resistance  is  the  inverse  slope  of  the 
/„,  1^-curve  and  is  given  by  equation  (15): 

r^^-Tr1  -  v     .....    (15) 


By  multiplying  numerator  and  denominator  by  the  expression  in 
the  parentheses  we  can  express  rp  in  the  simpler  form 


(40) 


or,  neglecting  the  small  quantity  e  and  putting  Eg  =  0, 


We  can,  therefore,  obtain  a  fair  estimate  of  the  plate  resistance  by 
simply  observing  the  plate  current  for  the  plate  voltage  at  which 
it  is  desired  to  obtain  the  resistance.  It  will  be  noted  that  the 
plate  resistance,  by  which  we  mean  the  a-c.  resistance,  is  half 
the  d-c.  resistance.  It  is  also  to  be  noted  that  while  the  amplifica- 
tion constant  /z  is  a  geometrical  constant,  the  plate  resistance 
depends  not  only  on  the  structure  of  the  tube  but  also  on  the  values 
of  the  plate  and  grid  voltages.  If  can,  however,  be  fully  specified 
for  all  operating  plate  and  grid  voltages  by  determining  it  as  a 
function  of  the  plate  voltage,  the  grid  voltage  being  kept  zero. 
The  relation  between  rp  and  Ep  can  be  represented  by  a  curve 
like  that  shown  in  Fig.  98.  Now,  the  resistance  at  any  specified 
plate  voltage  Ep  and  a  grid  voltage  Eg  other  than  zero  can  be 
obtained  by  applying  the  stray  field  relation  given  in  equation  (1), 
from  which  it  follows  (neglecting  e)  that  the  effective  plate  voltage 


196 


THERMIONIC  VACUUM  TUBE 


is  now  Ep+v.Eg.1  All  that  is  necessary,  therefore,  to  obtain  the 
plate  resistance  from  the  curve  in  Fig.  98  for  any  values  Ep  and  Eg 
is  to  read  off  the  resistance  at  an  abscissa  equal  to  Ep+vE0. 

In  regard  to  Fig.  98  it  should  be  noted  that  the  resistance  char- 
acteristic drops  in  virtue  of  the  increase  in  slope  of  the  plate  current 
characteristic.  Let  us  consider  the  curve  shown  in  Fig.  99,  which 
represents  the  relation  between  the  plate  current  and  the  effective 
plate  voltage  Ev  =  (Ep-\-^Eg).  If  this  voltage  has  the  value 
given  by  ob  the  direct  current  in  the  plate  circuit  will  be  repre- 
sented by  bb'.  Let  the  grid  voltage  now  be  varied  so  that  Ev 
oscillates  between  oa  and  oc,  ab  being  equal  to  be.  The  plate 
resistance  is  then  the  reciprocal  of  the  slope  of  the  line  a'c',  and  if 


Effective  Plate  Voltage 
FIG.  98. 

the  characteristic  is  parabolic  it  follows  directly  from  the  proper- 
ties of  the  parabola  that  a'c'  is  parallel  to  slope  of  the  curve  at  the 
point  corresponding  to  the  direct  voltage  Eir  =  ob.  In  the  case 
of  the  parabolic  characteristic  the  plate  resistance  is  therefore 
simply  given  by  the  slope  of  the  characteristic.  If  Ev  oscillates 
between  oc  and  od  the  plate  resistance  is  smaller  since  the  slope 
of  c'd'  is  larger.  If  now  Er  is  so  large  that  it  oscillates  between 
od  and  of  the  resistance  increases.  This  is  shown  by  the  broken 
part  of  the  resistance  characteristic  in  Fig.  98.  In  this  case  the 
resistance  is  no  longer  given  by  the  slope  of  the  curve  at  the 
point  corresponding  to  the  mean  value  of  Ev.  If  Ev  oscillates 
over  the  whole  range  oe  the  resistance  is  greater  than  in  the  case 

1  This  applies  for  positive  values  of  Eg  only  as  long  as  the  grid  is  not  suf- 
ficiently positive  to  take  an  appreciable  current. 


THE  THERMIONIC  AMPLIFIER 


197 


where  Ev  oscillates  over  the  range  cd  and  the  amplification  will 
be  less.  This  drop  in  amplification  when  the  input  becomes  very 
large  can  of  course  always  be  avoided  by  operating  at  a  higher 
plate  potential  Ep  and  increasing  the  saturation  current  by  in- 
creasing the  temperature  of  the  filament. 

Methods  have  been  devised  whereby  the  plate  resistance  can 
be  measured  dynamically  with  comparative  ease.     It  is  therefore 


a       o         c  cf  e 

Effective  Plate  Voltage 

FIG.  99. 


a  simple  matter  to  obtain  a  curve  like  that  shown  in  Fig.  98. 
The  following  method  was  published  by  J.  M.  Miller.1  Consider 
the  circuit  shown  in  Fig.  100.  It  will  be  recognized  that  with 
the  key  K\  open  and  Ki  closed  the  circuit  is  the  same  as  Fig.  97 
except  that  the  meter  is  replaced  by  the  telephone  receiver  T 
and  a  source  S  of  alternating  current  is  used  instead  of  the  battery 

1  Loc.  cit, 


198 


THERMIONIC  VACUUM  TUBE 


EI.  The  circuit  therefore  gives  a  means  of  measuring  /*,  which  can 
be  done  by  adjusting  r\  until  the  tone  in  the  receiver  T  vanishes. 
To  measure  the  plate  resistance  rp  let  the  key  K\  be  closed.  If 
eg  be  the  alternating  voltage  applied  between  filament  and  grid 
the  alternating  current  in  the  circuit  FPrq  is,  by  equation  (22), 


**  ° 


and  the  voltage  in  7*0  is  therefore 


M  9  ° 


Now  it  will  be 


observed  that  if  A  is  positive  and  B  negative,  the  electron  current 
to  the  plate  will  be  increased  if  the  effect  of  the  applied  grid 
voltage  exceeds  the  opposite  effect  of  the  voltage  simultaneously 
applied  to  the  plate.  The  currents  in  TQ  and  r\  are  therefore  in 
phase.  Hence,  by  adjusting  r0  until  the  potential  drop  in  it  is  equal 


A  ~— ^MA/WV 


FIG.  100. 

to  that  in  n,  the  tone  in  the  telephone  receiver  can  be  reduced  to  a 
minimum.     If  this  is  the  case  we  have 


But  eff  =  Ir2}  hence  the  plate  resistance  is: 


(42) 


from  which  rp  can  be  computed.  Miller  puts  ri=r2  to  obtain  a 
simpler  equation.  But  even  so  the  method  involves  a  calculation. 
A  very  valuable  simplification  hitherto  unpublished  was  suggested 
by  G.  H.  Stevenson.  Suppose  we  adjust  n  for  minimum  tone  in 


THE  THERMIONIC  AMPLIFIER  199 

T  when  KI  is  open.     Then  M  =  — ,  and  it  will  be  seen  from  equation 

(42)  that  with  this  relation  between  r\  and  7*2  it  would  not  be 
possible  to  obtain  a  balance  with  KI  closed.  But  if  r%  be  doubled, 
which  can  be  done  by  opening  K^  thus  adding  a  resistance  equal  to 
T2,  and  7*0  be  now  adjusted,  with  K\  closed,  to  give  minimum  tone 
in  T7,  then  rp  =  r$.  This  is  the  simplest  method  of  measuring  the 
plate  resistance.  By  giving  7*2  a  fixed  value  of,  say,  10  ohms  and 
calibrating  n  in  the  manner  explained  with  reference  to  Fig.  97, 
we  obtain  a  comparatively  simple  circuit  which  enables  us  to  read 
the  amplification  constant  and  the  plate  resistance  directly  in 
terms  of  r\  and  ro. 

67.  Direct  Measurement  of  the  Mutual  Conductance.  Once 
the  amplification  constant  and  plate  resistance  are  known  the 
mutual  conductance  can  be  obtained  from  equation  (18) 


(18) 


and  it  is  therefore  hardly  necessary  to  measure  it  directly.  How- 
ever, since  the  mutual  conductance  is  a  good  indication  of  the 
figure  of  merit  of  a  tube,  we  shall  briefly  describe  a  few  methods 
whereby  it  can  be  measured  directly.  Referring  to  equation 
(17)  it  can  be  seen  that  the  principle  of  any  method  of  direct  meas- 
urement of  the  mutual  conductance  is  to  apply  a  potential  differ- 
ence between  filament  and  grid  by  passing  a  current  through  a 
resistance  shunting  the  grid  and  filament  and  balancing  this  cur- 
rent against  the  resulting  current  in  the  plate  circuit.  There  are 
various  ways  in  which  this  can  be  done.  The  circuit  arrangement 
of  a  method  proposed  by  S.  Ballantine  l  is  shown  in  Fig.  101. 

The  coils  1  and  2  are  so  connected  that  the  currents  i\  and  i% 
flowing  in  the  directions  of  the  arrows  tend  to  neutralize  each 
other's  effect  in  the  secondary  of  transformer  T7.,  If  t\  and  fa 
be  the  inductance  due  to  the  coils  1  and  2,  respectively,  and  Ri  be 
so  adjusted  that  the  tone  in  the  receiver  is  a  minimum,  then 

*  1^1  =$2(2. 

Ballantine  assumes  that  12  =  —  °  from  which  he  then  obtains, 

rP 

1  S.  BALLANTINE,  Proc.  I.R.E.,  Vol.  7,  p.  134,  1919. 


200  THERMIONIC  VACUUM  TUBE 

since  eg=iiRi, 


(42) 


The  assumption  made  is,  strictly  speaking,  justifiable  only  when 
the  impressed  oscillations  are  very  small,  because  the  current- 
voltage  characteristic  of  the  circuit  is  not  linear  unless  the  external 
impedance  in  the  plate  circuit  (i.e.,  the  impedance  of  coil  2)  is 
large.  On  the  other  hand,  if  it  is  large  the  current  i2  cannot  be 


expressed  by  the  above  simple  equation,  but  is  given  by  equation 
(22),  namely: 


rP+Z2 


(22) 


where  Z2  is  the  impedance  of  coil  2.     The  mutual  conductance 
of  the  tube  is  then  given  by : 


that  is,  by 


ti     Z2 

. 

t2          fJL 


(43) 


(44) 


For  a  simplification  of  this  dynamic  method  I  am  indebted  to 
an  hitherto  unpublished  suggestion  of  Mr.  H.  W.  Everitt  which  is 


THE  THERMIONIC  AMPLIFIER 


201 


shown  in  Fig.  102.1  It  consists  in  replacing  the  transformer  T  by 
two  non-inductive  resistances  r\  and  r^  the  telephone  receiver 
being  connected  directly  to  them  as  shown.  The  effect  of  the 


FIG.  102. 

external  resistance  is  shown  in  the  following  table,  which  gives 
observations  obtained  by  Everitt.  The  second  column  gives  the 
values  that  would  be  obtained  with  equation  (42),  and  the  last 


1 

r2 

r\ 

TzRl       TZ 

100 
1,000 

1.33X10-3 
1.09X10-3 

1.37X10-3 
1.39X10-3 

2,100 

0.86X10-3 

1.34X10-3 

10,000 

0.37X10-3 

1.43X10-3 

column  the  corrected  values  according  to  equation   (44).     The 

true  value  of  —  computed  from  separately  observed  values  of  /x  and 
rp 

rp  is,  for  the  plate  voltage  used  in  these  experiments,  1.31X10"3. 
The  values  in  the  last  column  are  not  quite  in  agreement  with  this 
value,  since  the  method  is  not  very  accurate,  but  they  are  grouped 
around  a  mean  value.  The  values  given  in  the  second  column  are 
distinctly  influenced  by  the  external  resistance,  the  deviation  from 

1  This  modification  was  also  suggested  by  Ballantine  and  given  in  an  addeni- 
dum  to  his  paper,  which  appeared  about  four  months  after  the  reading  of  the 
original  paper  at  a  meeting  of  the  Institute  of  Radio  Engineers. 


202 


THERMIONIC  VACUUM  TUBE 


the  true  value  increasing  with  it.     When  the  external  resistance  is 
so  large  that  it  must  be  taken  into  consideration  the  method 

becomes  tedious  and  has  no  advantage  over  obtaining  -   from 

rP 

separate  determinations  of  ju  and  rP  by  the  method  explained  with 
reference  to  Fig.  100. 

A  simple  d-c.  method  of  measuring  the  mutual  conductance, 
due  to  E.  V.  Appleton  l  is  of  interest.  The  circuit  arrangement  is 
shown  in  Fig.  103.  When  the  key  K  is  open  the  galvanometer  G 
indicates  the  normal  plate  current.  When  K  is  closed  the  poten- 
tial difference  fy  applied  between  filament  and  grid  is  I\R\,  where 


FIG.  103. 


This  causes  a  change  in  the  plate  current 


7i  is  the  current  in  R\. 
equal  to 


Now  it  will  be  seen  that  /  and  I\  flow  through  the  galvanometer  in 
opposite  directions.  Hence  if  R\  be  adjusted  until  the  galvano- 
meter reading  shows  no  change,  then 


or 


_          _ 
Seg~Ri 


(45) 


The  plate  circuit  does  not  contain  any  external  resistance, 
except  that  of  the  galvanometer,  which  is  small.     This  equation  is, 

1  Wireless  World,  Vol.  6,  p.  458,  1918. 


THE  THERMIONIC  AMPLIFIER 


203 


strictly  speaking,  correct  only  when  the  potential  applied  to  the 

grid  is  small,  in  which  case  we  can  put  —  =  —  so  that 

5eg    rv 


(46) 


It  will  be  evident  from  the 
foregoing  that  the  mutual  con- 
ductance is,  like  the  plate  resist- 
ance, a  function  of  the  d-c.  plate 
and  grid  voltages.  As  in  the 
case  of  the  plate  resistance  the 
effect  of  the  plate  and  grid 
voltages  can  be  explained  with 
reference  to  a  curve  like  that 
shown  in  Fig.  99,  except  that 
here  the  abscissae  would  repre- 
sent the  effective  grid  voltage, 


E=   ^ °. 


effective 


instead    of    the 


plate     voltage,    Ev= 
It  is    evident  that 


68.  Circuit  for  Measuring 
Amplification  Constant.  Plate 
Resistance  and  Mutual  Conduc- 
tance. A  set  which  makes  possi- 
ble the  quick  measurement  of 
all  three  quantities,  ju,  rp  and  gm 
was  devised  by  H.  W.  Everitt. 
It  consists  of  the  combination 
of  three  circuits  shown  in  Fig. 
104.  For  a  certain  setting  of 
the  keys  on  the  box  the  circuit 
arrangement  is  that  shown  by 
circuit  I  (Fig.  104).  As  was 

explained  above,  when  r<2  is  so  adjusted  that  the  tone  in  the 
receiver  is  a  minimum,  then 


Tel,  Rec. 

FIG  104. 


7*2 


(47) 


204  THERMIONIC  VACUUM  TUBE 

The  resistance  r\  has  a  constant  value  of  10  ohms  and  r2  is  cali- 
brated to  read  tenths  of  ohms,  so  that  the  reading  of  r2  gives  /* 
directly. 

Now,  to  measure  rp  the  circuit  is  transformed  into  circuit  II 
by  the  simple  operation  of  throwing  over  a  multiple-contact  key. 
This  is  done  without  changing  the  setting  of  r2  that  gave  the  value 
of  n  in  circuit  I.  It  is  seen  that  r2  is  now  transferred  to  the  grid 
circuit  and  is  replaced  by  a  constant  resistance  rs  =  1000  ohms. 

Referring  to  this  circuit  and  applying  equations  given  in  the 
previous  sections,  the  voltage  drop  across  r^  is 

es  = 
If  ii  is  the  current  in  the  grid  circuit, 

Hence 

es  = 

When  R i  is  so  adjusted  that  the  tone  in  the  receiver  is  a  minimum, 
the  voltage  drop  e3  in  r3  is  equal  to  the  voltage  drop  i\r2  in  the 
resistance  r2.  Hence,  putting  63 =iir2  we  get: 


Now  since  r2  has  the  same  value  that  it  had  in  circuit  /  and 
ri  =  10  ohms,  we  have  from  equation  (47) 

r2  =  fj.ri  =  10/z. 
Hence 


""  3* 


10        10 

But  since  rs  is  a  constant  resistance  of  10  ohms  the  last  two 
terms  vanish.  Furthermore,  rs  =  1000  ohms,  so  that  the  plate 
resistance  is  given  directly  by  v 

rp=  lOOfli (48) 

The  dials  of  Ri  are  marked  100  times  their  actual  ohmic  resistance 
so  as  to  make  the  set  direct  reading. 

Next,  to  measure  the  mutual  conductance  gm  =  —  a  second 


THE  THERMIONIC  AMPLIFIER  205 

multiple-contact  key  is  operated  which  transforms  the  circuit 
into  III  (Fig.  104).  The  resistance  r%  is  the  same  as  that  used 
in  circuits  I  and  II. 

It  was  shown  in  Section  67  that  if  r±  is  small  compared  with  the 
plate  resistance  of  the  tube  then 


By  making  r^Rz  an  even  multiple  of  10,  we  have 


- 
rp 


For  the  chosen  values  of  r±  and  #2,  namely  100  and  1000  ohms, 
n  =  —  5,  and  since  the  dials  of  r^  are  marked  in  tenths  of  ohms  : 


(50) 


[r2\  being  the  reading  indicated  on  the  dials. 

The  measurement  of  the  tube  constants  with  this  set  is  a  very 
quick  and  simple  operation.  The  complete  set  is  shown  in  Fig. 
105.  It  includes  a  tone  source,  such  as  that  described  on  page  223. 
The  tube  to  be  tested  is  inserted  in  the  socket  as  indicated.  The 
transformation  of  the  circuits  is  accomplished  with  the  keys 
2  and  3.  A  and  B  represent  the  resistances  r^  and  R\  of  Fig.  104. 

69.  Influence  of  the  Electrode  Capacities.  The  amplification 
equations  derived  in  sections  61  to  63  express-  the  quantities 
considered  in  terms  of  the  potential  variations  actually  applied 
to  the  grid.  When  considering  the  power  supplied  to  the  input 
circuit  it  is  necessary  to  determine  to  what  extent  the  electrode 
capacities  can  influence  the  results.  The  potential  variations 
impressed  on  the  grid  when  the  power  is  supplied  to  the  input 
circuit  can  be  influenced  by  the  electrostatic  capacities  between 
the  electrodes  of  the  tube.  The  capacities  between  grid  and 
plate  effects  a  coupling  -  between  the  output  and  input  circuits, 
so  that  the  tube  is  not  a  perfect  unilateral  device.  The  extent 
to  which  the  output  circuit  reacts  on  the  input  depends  on  the 
constants  of  the  circuits. 

The  solution  of  the  network  involving  the  electrode  capacities 
was  given  by  H.  W.  Nichols  l  and  by  J.  M.  Miller.2 

1  H.  W.  NICHOLS,  Phys.  Rev.,  Vol.  13,  p.  405,  1919. 

2  J.  M.  MILLER,  Bureau  of  Standards,  Bulletin  No.  351. 


THERMIONIC  VACUUM  TUBE 


Fig.  106  represents  the  equivalent  network  of  the  tube  and  cir- 
cuit. (7,  F  and  P  denote  the  grid,  filament  and  plate.  This  cir- 
cuit represents  the  condition  that  the  grid  is  kept  at  a  negative 
potential  with  respect  to  the  filament,  so  that  there  is  no  con- 
vection current  between  them.  The  resistance  to  the  convection 
current  between  filament  and  plate  is  represented  by  rp  and  is  in 


FIG.  105. 

shunt  with  the  capacity  C2  between  the  filament  and  plate.  Zff 
represents  the  impedance  as  measured  between  filament  and  grid 
and  is  the  effective  input  impedance.  Remembering  that  a  poten- 
tial ea  impressed  on  the  grid  introduces  an  E.M.F.  equal  to  ney 
in  the  plate  circuit,  the  input  impedance  Zff  can  be  obtained  by 
including  in  the  plate  circuit  a  fictitious  generator  giving  ^ea  as 
indicated  in  the  diagram  and  solving  the  Kirchoff  equations  for  the 
network. 


THE  THERMIONIC  AMPLIFIER 


207 


Unless  the  frequency  is  very  high  (over  a  million  cycles  per 
second)  we  can  neglect  the  capacity  €2  between  filament  and 
plate,  since  it  is  shunted  by  the  plate  resistance  which,  is  then 
low  compared  with  the  impedance  due  to  €2.  The  equation  given 
by  Nichols  for  the  effective  input  impedance  is: 

7  =  _J 

"• 


where  W=    ^_£  ;    co  is  2?rX frequency,  and  ,;'  is  the  imaginary 
unit  V—  1.     The  other  quantities  are  indicated  in    Fig.   106. 

^v      * 

*»•: 


FIG.  106. 

For  most  tubes  used  at  present  this  equation  is  applicable  for 
frequencies  up  to  about  a  million  cycles  per  second. 

Let  the  external  output  impedance  take  the  general  form 
ZQ=rQ+jxQ.    Then  equation  (51)  can  be  transformed  into: 
_      ae+bd      ad-bc 
^  ' 


=  -rg+jxg 
where  the  coefficients  have  the  values 


(53) 
c  =  a)*rPro(J  iL 3  -1-  wx0(L  i  -t-  c 3  -t-M^ 3; 

It  will  be  seen  from  inspection  that  the  effective  input  im- 
pedance will  generally  comprise  a  resistance  rg  which  may  be 
positive  or  negative,  and  a  reactance  xg  which  is  capacitive. 

70.  Case  1.  Low  Frequencies:  co<106.  In  this  case  we 
can  neglect  co-terms  where  they  occur  in  the  same  expression  with 
terms  containing  co  in  a  lower  order,  e.g.,  neglect  co2  in  comparison 
with  co. 


208 


THERMIONIC  VACUUM  TUBE 


Let  the  output  impedance  be  inductive  XQ=LQU-     Evaluation 
of  the  coefficients  (53)  gives  for  the  input  resistance: 


and  for  the  input  reactance : 


and  therefore  the  effective  input  capacity  is: 


(55) 


(56) 


Now,  Ci  is  the  electrostatic  capacity  between  filament  and  grid. 
The  effective  input  capacity  is  greater  than  the  electrostatic 
capacity  by  the  amount  shown  by  equation  (56).  The  increase 
depends  on  the  electrostatic  capacity  between  grid  and  plate, 
and  on  the  resistance  in  the  output  load.  It  also  depends  en  the 
amplification  constant  //.  It  will  be  recognized  that  this  equation 
contains  the  expression  for  the  voltage  amplification  as  a  function 
of  the  external  output  resistance  (see  equation  28).  The  effective 
input  capacity  therefore  increases  with  the  output  lead  resist- 
ance in  the  manner  indicated  by  Fig.  95,  page  192.  Miller 1 
has  measured  the  effective  input  capacity  as  a  function  of  the 
external  output  resistance.  The  following  table  shows  the 
agreement  between  his  observed  and  computed  values  for  the  case 
of  a  VT-l  tube. 


Input  Capacity. 

r0,  Ohms. 

Computed. 

Observed. 

0 

27.9 

8,000 

51.4 

49.0 

16,000 

64.5 

61.5 

49,400 

78.9 

76.1 

97,000 

84.2 

84.3 

139,000 

86.1 

87.6 

1  Loc.  cit. 


THE  THERMIONIC  AMPLIFIER  209 

Equations  (54)  and  (55)  show  that  the  effective  input  resist- 
ance is  for  low  frequencies  («  <  106)  independent  of  the  frequency, 
but  depends,  among  the  other  circuit  constants,  on  both  resistance 
ro  and  inductance  LO  in  the  external  output  circuit.  The  input 
reactance,  on  the  other  hand,  is  inversely  proportional  to  the  fre- 
quency and  depends  on  the  resistance  TQ,  but  not  on  the  inductance 
in  the  output. 

From  equation  (55)  it  follows  that  the  amplification  given  by 
the  tube  would  decrease  as  the  frequency  is  increased.  But  this 
tendency  to  distort  is  in  itself  not  due  to  power  consumption  in 
the  input,  but  is  occasioned  by  the  decrease  in  the  input  grid 
potential,  due  to  the  lowering  of  the  input  reactance. 

The  power  consumed  in  the  input  is  determined  by  the  equa- 
tion (54).  This  equation  contains  a  negative  term  in  the  numer- 
ator and  therefore  if  the  output  inductance  LO  is  large  enough, 
the  input  resistance  can  be  negative.  Under  these  conditions  the 
tube  would  tend  to  produce  oscillations  or  "  sing  "  through  its 
internal  capacities.  This  tendency  to  sing  is  frequently  a  source 
of  annoyance  in  amplifier  circuits. 

Miller  has  computed  the  relation  between  rg  and  the  output 
inductance  LQ.  In  amplifier  circuits  we  are  usually  more  inter- 
ested in  the  external  output  impedance  than  in  the  output  induct- 
ance. Fig.  107  shows  the  relation  between  the  effective  input 

rp 

resistance  rQ  and  the  ratio  -  -  of  external  output  impedance  to 

rP 

plate  resistance  for  various  angles  0  =  tan~1  —  of  the  output 

ro 

impedance  Zo=ro+jLo.     For  a  pure  inductance  in  the  output 
(ro=0),  equation  (54),  reduces  to  the  simple  form 


_  0  (    . 

2' 


and  is  therefore  always  negative  .     As  the  angle  0  decreases,  the 
negative  value  of  rg  decreases  and  finally  becomes  positive. 

The  curves  in  Fig.   107  were  computed  with  the  following 
values  of  the  constants: 

rp  =  5Xl03  ohms 
Ci  =  5XlO-12  farad; 
C3  =  15  X10-12  farad; 

M  =  5; 

co  =  2X104. 


210 


THERMIONIC  VACUUM  TUBE 


These  are  approximately  the  constants  of  a  type  of  tube  that  is 
commonly  used  for  amplifying  telephonic  currents. 


FIG.  107. 

The  corresponding  input  reactances.  xa  are  shown  in  Fig.  108. 
The  input  reactance  is  therefore  generally  much  larger  than  the 
input  resistance. 


FIG.  108. 


2.5 


3.0 


3.5 


It  can  be  seen  from  the  above  equations  that  if  the  output 
reactance  is  capacitive,  then  the  input  resistance  is  always  positive 
This  is  also  the  case  when  the  output  is  a  pure  resistance  (zo=0). 


THE  THERMIONIC  AMPLIFIER  211 

Under  these  conditions  the  tube  would  absorb  power  from  the 
input.  But  for  ordinary  frequencies  this  power  absorption  is 
negligibly  small. 

The  power  absorbed  in  the  input  can  be  obtained  as  follows: 
Let  us  impress  an  alternating  potential  eg  on  the  grid.  Then  the 
grid  current  is 

*  =^£=_^_ 
Z.    rg+jxg> 

the  condition  being  that  the  grid  is  at  all  times  negative  with 
respect  to  the  filament,  so  that  there  is  no  convection  current 
between  filament  and  grid.  The  power  absorption  is  therefore  due 
entirely  to  the  reaction  of  the  output  on  the  input  circuit.  This 
power  is: 


or 

P  —  p  2 

f  a  —  Va 


For  co  <  106  we  can  neglect  rf  in  comparison  with  —97^-  and  write 

(58) 


Substituting  the  values  of  rff  and  Cg  from  equations  (54)  and 
(55),  for  the  case  of  a  pure  resistance  in  the  output  (Lo=0),  we 
obtain 

P9  =  <**e*rjCtfK(l+nK),     ....     (59) 

where    K  =  —  p—  .     This    shows    that    the    only    inter-electrode 


capacity  that  is  effective  in  causing  input  power  loss  is  Ca,  the 
capacity  between  grid  and  anode. 

In  order  to  obtain  the  order  of  magnitude  of  Pg  for  a  common 
type  of  tube,  we  can  insert  the  values  for  the  constants  given  on 
page  209  into  equation  (59).  When  ro=rp  (the  condition  for 
maximum  power  output),  K  =  %.  Putting  eg  =  5  volts,  we  get: 

P,  =  4.8X10-17Xco2  watt. 

For  a  tube  of  the  type  considered,  and  for  telephonic  fre- 
quencies (co<2X104),  the  ratio  of  output  to  total  input  power 


212  THERMIONIC  VACUUM  TUBE 

is  about  300,  and  for  an  input  voltage  of  5  volts  the  power  in  the 
output  is  about  30X10"3  watt.  The  total  input  power  is  there- 
fore about  1  X  10"4  watt.  For  co  =  2  X  104,  the  power  Pg  consumed 
in  the  effective  grid  resistance  is  about  2X10~8,  which  is  still 
very  small  compared  to  the  total  input  power  which,  in  normal 
operation  of  a  tube,  is  consumed  in  the  input  transformer,  and  in 
the  high  resistance  usually  bridged  across  filament  and  grid, 
as  shown  in  Fig.  90,  page  182.  Of  course,  the  power  consumption 
by  the  effective  grid  resistance  can  become  quite  large  when  the 
frequency  is  high,  since  it  increases  with  the  square  of  the  frequency 
within  the  frequency  range  considered.  At  extremely  high 
frequencies  the  effective  grid  resistance  again  becomes  negligibly 
small,  as  will  be  seen  from  the  following. 

71.  Case  II.  High  Frequencies.  When  the  frequency  is 
very  high,  we  cannot  neglect  the  capacity  €2  between  filament  and 
plate  because  the  impedance  due  to  €2  can  obviously  become  com- 
parable with  or  even  lower  than  the  plate  resistance  rp  with  which 
it  is  in  parallel.  (See  Fig.  106.)  In  this  case  the  effective  input 
impedance  is  given  by 

„  _ac-\-bd     .be—  ad 
°~^+d?^~J#+d?' 

where  the  coefficients  now  have  the  values: 


(60) 
and  xQ  can  be  L0to  or  ~  —  .     Since  co  is  large  we  can  neglect  the 


co-terms  of  lower  order  in  comparison  with  those  of  the  succeeding 
and  higher  orders.  This  gives: 

rg  =  Q  i 

r,  _<?iC3+CiC2+C2C3  [,  .....     (61) 
C2+C3  J 

independent  of  the  constants  of  the  external  output  circuit.  At 
very  high  frequencies,  therefore,  the  grid  does  not  absorb  power, 
but  the  amplification  is  lowered  because  the  input  current  is 
practically  short-circuited  by  the  electrode  capacities. 


THE  THERMIONIC  AMPLIFIER 


213 


The  voltage  amplification  as  a  function  of  the  frequency, 
for  frequencies  ranging  from'  3  X 105  to  4  million  cycles  per  second, 
can  be  measured  with  the  circuit  arrangement  shown  in  Fig.  109, 
for  which  I  am  indebted  to  my  associate  Dr.  J.  B.  Johnson.  A\ 
is  the  tube  with  which  the  high-frequency  current  was  amplified. 
The  a-c.  output  voltage  from  this  tube  was  measured  by  means 
of  a  bridge  arrangement  shown  on  the  right  of  the  figure,  in 
which  the  plate-filament  resistance  of  the  tube  A  formed  one 
arm  of  the  bridge.  The  other  three  arms  are  indicated  by  B,  C 
and  D.  The  output  from  the  tube  A\  is  impressed  on  the  grid 

p 

WVMW 


B 

A/WWW1 — 


FIG.  109. 

of  the  tube  A,  which  acts  as  a  detector,  so  that  small  potential 
variations  applied  to  its  grid  unbalances  the  bridge. 

The  bridge  was  calibrated  by  applying  known  a-c.  voltages 
to  the  grid  of  tube  A,  and  noting  the  deflection  of  the  galvanometer 
G.  The  known  voltage  is  applied,  as  is  commonly  done,  by  con- 
necting between  the  grid  and  filament  of  tube  A,  a  small  resistance 
and  a  thermocouple  in  series.  The  high-frequency  current  is 
passed  through  the  resistance  and  thermocouple,  and  the  input 
voltage  is  given  by  the  known  value  of  the  resistance,  and  the 
reading  is  indicated  by  the  thermocouple.  In  order  to  measure 
the  amplification,  the  tube  AI  is  inserted  as  shown,  and  the  input 
voltage  measured  with  resistance  and  thermocouple  as  indicated. 
In  such  case  we  therefore  measure  the  voltage  actually  applied  to 
the  grid  of  the  tube  AI,  so  that  the  capacity  between  filament  and 


214 


THERMIONIC  VACUUM  TUBE 


grid  of  this  tube  does  not  affect  the  results.  Now,  the  actual 
voltage  amplification  depends  on  the  load  in  the  output  of  the  tube. 
This  load  is  constituted  by  the  resistance  r0  and  n  in  parallel. 
This  is,  however,  not  the  only  impedance  in  the  output  of  tube  AI, 
because  there  is  parallel  with  r0  and  n,  the  bridge  circuit  connected 
through  the  capacities  of  tube  A,  which  forms  one  arm  of  the 
bridge.  Hence,  unless  these  capacities  are  very  small  the  values  of 
the  voltage  in  the  output  of  tube  AI  (and  which  is  impressed  on  the 
input  of  tube  A)  will  be  smaller  than  the  output  voltage  that  would 
actually  be  given  by  the  tube  AI  if  its  output  were  not  connected 
to  the  bridge.  The  galvanometer  would  therefore  indicate  read- 


il 

! 

fz 


M 


zoo 


300 


400  500          600 

Wave  Length -Meters 

FIG.  110. 


700 


900 


1000 


ings  that  are  too  small.  In  order  to  avoid  such  an  effect,  the  tube 
A  was  specially  constructed  to  have  as  small  a  capacity  as  possible 
between  its  electrodes.  This  capacity  should  be  small  compared 
with  the  capacity  between  the  elements  of  the  tube  AI.  Fig.  110 
shows  two  curves  obtained  by  Johnson  which  indicate  the  relation 
between  the  voltage  amplification  and  the  wave  length  for  a  stand- 
ard type  of  tube  such  as  is  shown  in  Fig.  68.  The  two  curves 
are  for  different  plate  potentials.  The  higher  plate  potential  of 
course  gives  a  higher  amplification  because  the  plate  resistance 
of  the  tube  is  lower.  It  is  seen  that  the  amplification  at  1000 
meters  is  about  three  times  as  large  as  the  amplification  at  100 
meters,  the  amplification  at  both  plate  potentials  dropping  to 
zero  when  the  frequency  comes  infinitely  high. 


THE  THERMIONIC  AMPLIFIER  215 

The  reduction  in  the  amplification  can  for  a  given  frequency 
be  avoided,  as  Nichols  suggested,  by  shunting  the  grid-plate 
capacity  with  a  inductance  of  such  a  value  as  to  make  the  im- 
pedance between  grid  and  plate  infinite  at  the  given  frequency. 
The  value  of  this  inductance  would,  of  course,  depend  on  the  other 
reactances  in  the  circuit.  Making  the  grid-plate  impedance 
infinite  is  equivalent  to  putting  Cs^Q  in  equation  (61).  Then 
Cg  =  Ci,  and  this  is  in  parallel  with  the  tuning  capacity  in  the 
input,  which  is  generally  shunted  across  the  input  inductance. 
By  properly  adjusting  the  tuning  capacity  the  filament-grid  im- 
pedance can  be  made  large,  thus  increasing  the  input  potential 
applied  to  the  grid.  This  scheme  has  now  been  in  use  for  several 
years  and  found  to  give  satisfactory  results. 

72.  Practical  Measurement  of  Amplification.  It  will  be 
evident  that  the  amplification  that  can  be  produced  by  a  tube 
depends  not  only  upon  its  structural  parameters,  but  also  upon 
the  constants  of  the  circuit  in  which  it  is  used.  When  a  tube  is 
designed  for  a  specific  purpose  and  is  to  be  operated  in  a  specific 
circuit,  the  amplification  can  be  measured  quickly  and  conveni- 
ently by  a  method  tliat  has  been  in  use  in  telephone  practice  for 
a  number  of  years.  The  desirability  of  a  method  of  testing  tubes 
rapidly  for  amplification  becomes  apparent  where  they  are  manu- 
factured in  comparatively  large  quantities.  These  tubes  are,  for 
example,  used  extensively  on  the  long  distance  telephone  lines 
of  the  Bell  System  and  elsewhere  as  telephone  repeaters.  Tubes 
used  for  this  purpose  are  made  with  great  care  and  are  designed 
to*  operate  in  circuits  of  definite  constants,  the  variation  of  the 
tube  constants  being  kept  within  close  limits. 

The  method  of  measuring  amplification  consists  in  amplifying 
a  current  of  a  given  frequency  with  the  tube  and  then  passing 
the  amplified  current  through  an  artificial  line  of  variable  and 
known  attenuation  before  it  passes  through  a  telephone  receiver. 
By  means  of  a  switch  the  current  can  also  be  transmitted  directly 
from  the  source  to  the  receiver.  When  the  attenuation  of  the 
artificial  line  is  so  adjusted  that  the  intensity  of  the  current  in 
the  receiver  is  the  same  whether  it  passes  directly  to  the  receiver 
or  through  the  tube  and  line  to  the  receiver,  then  the  current  is 
attenuated  by  the  line  as  much  as  it  is  amplified  by  the  tube,  and 
the  amplification  can  therefore  be  obtained  from  the  known  con- 
stants of  the  line. 


216 


THERMIONIC  VACUUM  TUBE 


The  circuit  arrangement  used  for  testing  audio  frequency 
amplifiers  is  shown  in  Fig.  111.  U  is  a  source  of  alternating 
current  which  will  be  described  later.  The  current  is  preferably 
passed  through  a  wave  filter  to  obtain  a  pure  note  of  about  800 
cycles  per  second.  By  means  of  the  switch  W  this  current  can  be 
transmitted  directly  to  the  telephone  receiver  or  be  impressed 
on  the  input  circuit  of  the  tube.  S  represents  the  artificial  line 
and  takes  the  form  of  a  shunt  to  the  receiver.  It  is,  however, 


FIG.  111. 

not  a  simple  shunt,  but  is  so  designed  that  the  total  impedance  to 
the  output  Current  is  always  constant  for  all  values  of  the  branch 
current  in  the  receiver.  This  is  done,  as  will  be  explained  below, 
by  the  addition  of  the  series  resistance  r\.  In  this  form  the 
shunt  is  known  as  a  "  receiver  shunt."  The  direct  current  in  the 
plate  circuit  is  supplied  through  the  choke  coil,  as  shown,  to 
insure  that  the  plate  resistance  remains  constant.  Although  the 
total  output  impedance  remains  constant  for  all  adjustments  of  the 
shunt  the  resistance  does  not.  Hence,  if  the  choke  coil  were 
omitted  the  potential  difference  between  filament  and  plate  would 
change  with  the  shunt  adjustment  and  this  would  result  in  a  change 


THE  THERMIONIC  AMPLIFIER  217 

in  the  plate  resistance.  The  secondary  of  transformer  T<z  is  wound 
to  have  as  high  an  impedance  as  possible,  so  as  to  impress  the 
highest  possible  voltage  between  the  grid  and  filament.  The 
grid  battery  Eg  is  inserted  to  keep  the  grid  sufficiently  negative 
with  respect  to  the  filament  to  prevent  it  from  taking  current. 
It  is  important  to  note  that  the  current  drawn  from  the  generator 
U  must  remain  the  same  for  both  positions  of  the  switch  W.  The 
primary  of  transformer  T%  is  therefore  wound  so  that  the  im- 
pedance as  measured  across  the  terminals  of  the  primary  is  the  same 
as  that  of  the  telephone  receiver. 

Now,  if  ip  be  the  total  alternating  current  in  the  plate  circuit, 
and  IQ  the  branch  current  in  the  receiver,  then  we  can  put 


(62) 


where  d  represents  the  length  of  artificial  line  of  which  a  is  the 
attenuation  constant  per  unit  length. 

The  value  of  d  depends  of  course  upon  the  relative  values  of 
the  series  and  shunt  resistances  r\  and  r%  of  the  receiver  shunt. 
If  these  are  so  adjusted  that  the  intensity  of  the  note  in  the  tele- 
phone receiver  is  the  same  for  both  positions  of  the  switch  W, 
then  the  expression  ead  gives  a  measure  of  the  current  amplifica- 
tion produced  by  the  tube  provided  the  following  conditions  are 
satisfied.  First,  the  current  io  in  the  telephone  receiver  must 
be  equal  to  the  current  in  the  primary  of  transformer  T%.  This 
can  be  done  by  making  the  impedance  Zi  of  T%  equal  to  that  of  the 
receiver  T.  Second,  for  "  amplification  "  to  have  any  meaning 
the  impedance  to  the  input  current  io  must  be  equal  to  the  im- 
pedance to  the  output  current  ip.  Now  the  total  impedance  to  the 
input  current  is  that  of  the  circuit  Zf\Z\  (which  is  equal  to  that  of 
the  circuit  Z\T)  and  the  impedance  to  the  output  current  is  that 
of  the  circuit  FPATB.  These  two  impedances  must  be  equal. 
The  receiver  shunt  must  therefore  be  so  designed  that  for  all  the 
necessary  adjustments  of  its  shunt  and  series  resistances  r^  and  r\ 
the  total  impedance  to  the  output  current  remains  constant. 
If  the  plate  resistance  differs  markedly  from  the  impedance  of  the 
receiver  a  transformer  can  be  inserted  between  the  coil  AB  and 
the  receiver  shunt  S. 

Writing  equation  (62)  in  the  form 

d  =  Klog,        ......     (63) 


218  THERMIONIC  VACUUM  TUBE 

where 


(64) 


we  can  express  the  current  amplification  in  terms  of  d  instead  of 

—  that  is,  we  express  it  on  the  logarithmic  scale. 
*o 

The  constant  a  is  arbitrary  and  can  be  given  any  convenient 
value,  provided  it  be  definitely  specified  when  the  amplification 
is  expressed  in  terms  of  d.  Unfortunately  the  value  of  a  commonly 
used  in  telephone  practice  is  not  a  convenient  value,  but  it  has 
already  found  its  way  into  extensive  vacuum  tube  practice  and 
we  shall  adopt  it  here.  As  commonly  used,  a  is  the  attenuation 
constant  per  mile  of  the  so-called  "  standard  No.  19  gauge  cable," 
which  has  a  capacity  of  0.054  mf.,  and  a  resistance  of  88  ohms  per 
mile.  It  is  to  be  noted  that  this  reference  cable  has  neither 
inductance  nor  leakance.  The  constant  a  is  determined  by  these 
two  quantities  and  the  frequency  of  the  current.  If  the  frequency 
is  800  cycles  per  second,  «  =  0.1091.  This  makes  K  =  21.13  and 
the  amplification  can  thus  be  expressed  in  terms  of  miles  (d) 
of  standard  cable.  And  this  simply  means  that  if  the  amplifica- 
tion is  d  miles,  it  would  take  a  standard  No.  19  gauge  cable  d  miles 
long  to  reduce  the  current  to  its  original  value.  When  expressing 
power  amplification  instead  of  current  amplification  the  constant 
K  in  equation  (63)  must  be  divided  by  two. 

There  is  an  advantage  in  expressing  amplification  on  the 
logarithmic  scale  which  is  in  accordance  with  our  lack  of  con- 
ception of  absolute  intensity  of  sound.  By  using  the  logarithmic 
scale  and  taking  length  of  cable  as  the  standard  of  reference,  we 
obtain  a  definite  idea  of  what  the  energy  means  when  it  is.  to  be 
used  for  operating  a  telephone  receiver.  The  value  of  the  attenua- 
tion constant  a  happens  to  be  such  that  steps  of  1  mile  of  the 
chosen  standard  cable  afford  a  very  convenient  unit  of  measure- 
ment. For  many  purposes  it  is,  however,  sufficient  if  the  receiver 
shunt  is  calibrated  in  steps  of  two  miles  of  the  standard  cable. 
A  little  practice  then  makes  it  possible  to  estimate  amplification 
to  within  1  mile. 

For  convenience  of  reference  the  following  table  is  attached, 
giving  the  relation  between  miles  of  the  standard  No.  19  gauge 
cable  and  the  current  and  power  ratios  as  computed  from  equations 
(63)  and  (64). 


THE  THERMIONIC  AMPLIFIER 


219 


Miles  of  Std. 
Cable. 

Current 
Ratio. 

Power 
Ratio. 

1 

1.11 

1.24 

2 

1.24 

1.54 

3 

1.38 

1.92  ' 

4 

1.54 

2.39 

5 

1.72 

2.97 

6 

1.92 

3.70 

7 

2.14 

4.60 

8 

2.39 

5.72 

9 

2.66 

7.12 

10 

2.97 

8.85 

11 

3.31 

11.02 

12 

3.70 

13.7 

13 

4.12 

17.0 

14 

4.49 

21.1 

15 

5.12 

26.37 

16 

5.71 

32.7 

17 

6.94 

40.7 

18 

7.11 

50.6 

19 

7.92 

62.8 

20 

8.83 

78.3 

21 

9.84 

97.4 

22 

11.0 

121.1 

23 

12.25 

150.7 

24 

13.65 

187.1 

25 

15.2 

233.4 

26 

17.0 

289.8 

27 

18.9 

359.8 

28 

21.1 

447.8 

29 

23.5 

559.2 

30 

26.3 

693.5 

32 

32.6 

1.07   X103 

34 

40.5 

1.65   XlO3 

36 

50.5 

2.57  XlO3 

38 

62.6 

3.96   XlO3 

40 

78.1 

6.15   XlO3 

42 

97.2 

9.44   XlO3 

44 

120.8 

1.47   XlO4 

46 

150.6 

2.27   XlO4 

48 

186.6 

3.51   XlO4 

50 

232.3 

5.43   XlO4 

55 

400.0 

1.61   XlO5 

60 

691.9 

4.79  XlO5 

65 

1.18X103 

1.43   XlO6 

70 

2.05X103 

4.  265  XlO6 

75 

3.54X103 

1.285  XlO7 

80 

6.09X103 

3.75  XlO7 

85 

10.55X103 

1.13  XlO8 

90 

18.16X103 

3.32  XlO8    . 

95 

31.27X103 

9.95  XlO8 

100 

53.83X103 

2.95  XlO9 

220 


THERMIONIC  VACUUM  TUBE 


Fig.  112  shows  the  relation  between  power  amplification  and 
the  ratio  of  external  impedance  ZQ  to  the  plate  resistance  rp. 
The  curves  were  computed  from  equation  (35)  with  the  following 
values :  /z  =  5,  rp  =  5  X 103  ohms,  rg  =  6  X 105  ohms,  and  the  imped- 
ance ZQ  was  assumed  to  have  an  angle  of  45°.  The  lower  curve 
gives  the  amplification  expressed  in  ratio  of  output  to  input  power, 
while  the  upper  curve  gives  the  power  amplification  expressed  in 
miles  of  standard  No.  19  gauge  cable.  It  is  seen  that  while  the 

power  ratio  varies  considerably  from  the  maximum  value  when  the 
n 

ratio  —  deviates  from  unity,  yet  the  change  in  the  effect  produced 
rp 

on  the  ear,  which  is  more  in  accordance  with  the  upper  curve 


ICO 


toe 


500 


400 


JOO 


27. 5 


2.0 


2.5 


3.0 


FIG.  112. 

expressed  on  the  logarithmic  scale,  is  quite  small.  A  change 
in  amplification  of  one  standard  cable  mile  is  not  serious.  The 
external  impedance  into  which  the  tube  works  can  therefore  have 
values  ranging  from  about  one-half  to  two  and  one-half  times  the 
plate  resistance  without  producing  any  marked  change  in  effect  as 
heard  in  the  telephone  receiver. 

Now,  it  was  stated  above  that  the  condition  to  be  satisfied  by 
the  receiver  shunt  is  that  its  insertion  in  the  circuit  must  not  change 
the  total  impedance  of  that  circuit,  and  this  must  be  true  for  all 
adjustments  of  its  shunt  and  series  resistances.  The  receiver 
shunt  is,  of  course,  so  arranged  that  the  shunt  and  series  resist- 
ances are  only  inserted  in  definite  pairs,  the  object  being  that  the 
change  in  the  total  impedance,  due  to  the  insertion  of  a  shunt 


THE  THERMIONIC  AMPLIFIER 


221 


resistance  must  be  compensated  for  by  the  addition  of  a  corre- 
sponding series  resistance. 

Referring  to  Fig.  113,  let  n,  r^  represent  the  receiver  shunt, 
rp  the  plate  resistance  of  the  tube  which  is  non-reactive  and  con- 
stant, and  ZQ  the  impedance  of  the  receiver,  the  angle  of  which  is 

tan"1  — ,  and  this  is  the  angle  of  the  external  impedance  when  the 

ro 

receiver  shunt  is  cut  out  (ri  =  0,  7*2  =  oo ).  But  when  the  shunt  is 
inserted  the  angle  of  the  effective  external  impedance  depends 
on  the  values  of  ri  and  r%,  and  changes  with  every  adjustment 
of  7*1  and  T2.  When  the  angle  of  the  receiver  is  large  and  if 
accuracy  is  desired,  this  must  be  taken  into  consideration  in  com- 


FIG.  113. 

puting  receiver  shunts.  The  necessary  values  of  r\  and  7*2  can  be 
determined  graphically  or  in  the  following  simple  manner  by  first 
neglecting  the  effect  of  the  change  in  angle  and  then  applying  a 
simple  correction  method. 

The  first  condition  to  be  satisfied  is  that  the  current  ^o  in  the 
impedance  ZQ  must  be  related  to  the  total  output  current  ip  in 
the  plate  circuit  by  the  equation 


(62) 


where  the  current  attenuation  produced  by  the  shunt  is  then 
equivalent  to  d  miles  of  cable  of  which  a  is  the  attenuation  constant 
per  mile  at  the  frequency  of  the  current,  which  we  shall  take  to  be 
800  cycles  per  second,  thus  making  a  =  0.109  for  the  "  standard 
cable." 

Now  the  alternating  current  established  in  the  circuit  (Fig.  113) 
is  due  to  the  alternating  potential  eg  applied  to  the  grid.     The 


222  THERMIONIC  VACUUM  TUBE 

impressed  E.M.F.  is  therefore  ^eg  and  is  applied  in  the  branch 


If  Z  be  the  total  impedance  of  the  circuit  and  R  and  X  the 
resistance  and  reactance,  respectively,  of  the  circuit  to  the  right 
of  A  B,  we  have  as  a  second  condition  to  be  satisfied  by  the  receiver 
shunt: 


constant.     .     .     .     (65) 

v 

When  the  shunt  is  cut  out  (n  =  0,  r2  =  <*>  )  then  R  =  r  0  and  X  =  x0. 
Summing  the  E.M.F.'s  in  the  two  branches  we  have 

.....     (66) 
'  •     .   '(67) 
From  (67)  and  (62)  we  obtain  directly 


or 

r2=-r         ?  ,Z°         .     ,.     .     .     .  (68) 

smh  ad  +  cosh  ad  —  1 

The  values  of  r^  necessary  to  give  any  desired  attenuation  d  can 
thus  be  obtained  directly  from  a  table  of  hyperbolic  functions. 

In  deriving  equation  (68)  we  made  use  of  equation  (62), 
which  holds  for  a  circuit  of  zero  reactance,  while  in  the  receiver 
shunt  circuit  the  reactance  is  not  zero.  The  values  of  r%  obtained 
will  therefore  not  be  correct  unless  the  angle  of  the  impedance  ZQ 
is  small  or  .the  attenuation  large.  If  the  angle  of  ZQ  is  not  greater 
than  about  45°  the  values  of  TI  given  by  equation  (68)  are  suf- 
ficiently accurate  for  current  attenuations  greater  than  those 
produced  by  6  miles  of  standard  cable  (d>6).  For  smaller  atten- 
uations or  if  the  impedance  ZQ  has  a  large  angle  the  values  of 
r2  obtained  from  (68)  can  be  corrected  as  follows:  Suppose  it 
is  desired  to  compute  a  receiver  shunt  giving  a  maximum  attenua- 
tion of  30  miles  of  standard  cable  and  allowing  the  attenuation  to 
be  varied  in  steps  of  1  mile  each.  This  shunt,  we  shall  suppose, 
is  to  operate  with  a  receiver  having  a  large  angle,  say,  70°.  We 
can  use  equation  (68)  to  obtain  an  idea  of  the  range  of  values 
of  r<i  that  would  be  necessary  to  give  attenuations  ranging  from 
1  to  30  miles,  by  computing  r<i  for  d  equal  to  2  and  30,  say.  By 
choosing  five  or  six  convenient  arbitrary  values  of  r%  covering 


THE  THERMIONIC  AMPLIFIER 


223 


this  range,  we  can  compute  the  corresponding  current  attenua- 
tions -7^,  or  d,  that  would  be  produced  by  the  chosen  values  of  7*2 

IQ 
in  parallel  with  ZQ=TQ-\-JXQ,  and  by  plotting  a   curve  between 

these  values  of  d  and  the  chosen  values  of  r%,  the  correct  shunt 
resistances  can  be  read  from  the  curve  for  all  the  desired  values 
of  d  from  1  to  30  miles.  It  will  be  recognized  that  by  this  method 
the  shunt  is  computed  by  the  quick  and  simple  process  of  deter- 
mining d  for  chosen  values  of  TI  instead  of  the  lengthy  and  tedious 
way  of  determining  r^  from  the  desired  values  of  d. 

Once  the  values  of  the  shunt  resistances  r^  are  known,  the 
corresponding  series  resistances  r\  can 
be  obtained  from  the  condition  stated 
by  equation   (65)   that  the  total  im- 
pedance must  remain  constant. 

The  generator  U  (Fig.  Ill)  can 
be  of  any  type  that  gives  a  constant 
note  of  about  800  cycles  per  second. 
It  may,  for  example,  be  an  audion 
oscillator  or  a  microphone  generator, 
as  shown  in  Fig.  111.  This  type  of 
generator  is  very  convenient  when 
compactness  is  desired.  Its  principle 
of  operation  is  like  that  of  an  inter- 
rupter or  "  buzzer,"  although  it  is 
much  superior  to  the  interrupter, 
which  is  for  this  purpose  practically 
useless  because  of  its  inconstancy  and 
need  for  constant  adjustment.  The 

microphone  generator  is  shown  diagrammatically  in  Fig.  114. 
C  is  a  carbon  button,  the  diaphragm  d  of  which  is  under  tension, 
due  to  the  pressure  of  the  armature  a  against  the  pin  p.  The 
current  from  the  battery  causes  the  armature  to  be  attracted  to 
the  coil.  This  releases  the  pressure  on  the  diaphragm  and  in- 
creases the  resistance  cf  the  carbon.  The  resulting  decrease  in 
current  reverses  the  process  and  an  alternating  current  is  estab- 
lished in  a  circuit  connected  to  AB.  It  is  seen  that  the  current 
is  never  broken  as  in  the  case  of  an  interrupter,  but  the  current 
strength  is  merely  varied  by  the  varying  pressure  exerted  on  the 
carbon.  This  device  is  therefore  free  from  troubles  attending 


FIG.  114. 


224 


THERMIONIC  VACUUM  TUBE 


the  interrupter,  such  as  corroding  of  contacts,  etc.  Fig.  115 
shows  a  photograph  of  the  microphone  generator.  It  operates  on 
a  voltage  of  3  to  5  volts  and  gives  an  alternating  current  output 
of  several  milliamperes  in  a  resistance  equal  to  its  own  resistance 
which  varies  approximately  between  50  to  100  ohms. 

The  use  of  this  generator  makes  it  possible  to  include  the  whole 
amplification  test  circuit  shown  in  Fig.  Ill,  in  compact  form  in  a 
portable  box.  Fig.  116  shows  such  an  amplifier  test  set  used  for 
measuring  the  amplification  of  tubes  in  the  laboratories  of  the 
Western  Electric  Company.  The  dial  of  the  receiver  shunt 
is  shown  at  the  lower  right-hand  corner  of  the  box. 


FIG.  115 

73.  Amplification  as  a  Function  of  the  Operating  Parameters. 
In  the  early  part  of  this  chapter  it  was  pointed  out  that  the 
operating  range  of  the  characteristic  of  the  tube  is  characterized 
by  the  condition  that  the  filament  temperature  must  be  so  high 

that  the  effective  grid  voltage  ( —  -\-Eff-\-e )  is  not  high  enough  to 

draw  all  the  electrons  to  the  anode  as  fast  as  they  are  emitted  from 
the  cathode,  for  if  this  were  so,  a  change  in  the  applied  voltage 
would  not  produce  any  appreciable  change  in  the  anode  current. 
In  other  words,  the  temperature  of  the  filament  must  be  high 
enough  to  comply  with  the  condition  of  "  temperature  saturation  " 


THE  THERMIONIC  AMPLIFIER 


225 


which  is  indicated  by  the  horizontal  part  of  the  plate  voltage- 
filament  current  characteristic  (see  Fig.  18,  page  51).  The  effect 
of  filament  temperature  is  indicated  in  Fig.  117,  which  gives  the 
relation  between  the  filament  voltage  and  the  amplification  meas- 
ured at  a  constant  plate  voltage  in  a  circuit  like  that  shown  in 
Fig.  111.  For  this  particular  type  of  tube,  it  is  seen  from  the 
curve,  the  filament  voltage  must  never  drop  below  3.0  volts.  On 
the  other  hand,  it  should  not  be  increased  more  than  is  absolutely 
necessary ,  for  this  would  shorten  the  life  of  the  filament.  Satisf  ac- 


FIG.  116. 

tory  operation  and  long  life  can  therefore  be  obtained  by  suitably 
adjusting  the  filament  current. 

Referring  to  equation  (36)  it  is  evident  that  the  amplification 

2 

increases  as  the  ratio  —  is  increased  provided  the  plate  resistance 

Tp 

rp  always  remains  equal  to  the  external  impedance  of  the  plate 
circuit.  For  a  given  type  of  tube  M  remains  practically  constant, 
but  rp  can  be  decreased  by  increasing  the  plate  potential  provided 
the  filament  temperature  remains  high  enough  to  insure  "  tempera- 
ture saturation."  (It  will  be  remembered  that  the  filament 
temperature  necessary  for  this  condition  increases  as  the  plate 
voltage  is  increased.)  It  is  therefore  to  be  expected  that  the 


226 


THERMIONIC  VACUUM  TUBE 


amplification  would  increase  with  increase  in  the  plate  voltage. 
If  the  amplification  is  measured  for  increasing  plate  voltages  Ep 
in  the  circuit  of  Fig.  Ill,  which  contains  a  fixed  external  impedance, 
the  amplification  generally  tends  toward  a  maximum  value,  as 
shown  in  Fig.  118.  This  is  due  to  the  counter  balancing  effect 

rj 

occasioned  by  the  deviation  of  the  ratio  —  from  unity.     (See 

rp 

Fig.  112.^ 


Filament  Volts 
FIG.  117. 

74.  Tube  Constants  as  Functions  of  the  Structural  Parameters. 
It  will  be  recognized  from  the  foregoing  that  the  amplification 
constant  M  and  the  plate  resistance  rp  play  an  important  part 
in  the  operation  of  three-electrode  tubes.  They  are  the  two  main 
constants  that  figure  in  the  design  of  such  tubes.  Except  at  low 
effective  voltages,  ju  is  practically  independent  of  the  applied 
grid  and  plate  potentials,  and  is  determined  by  the  dimensions  of 
the  grid  and  the  distance  between  grid  and  plate.  The  plate 
resistance  depends  not  only  on  the  dimensions  and  disposition  of 
the  electrodes,  but  also  on  the  applied  voltages.  It  is  important 
to  know  for  the  purpose  of  designing  tubes,  how  these  constants 
depend  on  the  structural  dimensions  of  the  tube. 


THE  THERMIONIC  AMPLIFIER 


227 


75.  Calculation  of  Amplification  Constant.  The  constant  M  is 
due  to  the  electrostatic  screening  effect  of  the  grid.  An  expression 
for  this  effect  had  been  derived  by  Maxwell  long  before  the 
audion  came  into  existence.1  The  problem  that  Maxwell  set 
himself  was  to  determine  the  extent  to  which  a  wire  grating  or 
gauze  could  protect  apparatus  enclosed  by  it  from  external  elec- 
trostatic disturbances.  His  solution  is,  however,  directly  applic- 
able to  the  audion.  It  was  applied  and  extended  to  include 
cylindrical  tubes  by  Abraham,  King,  Schottky,  and  v.  Laue.2 


I  15 


'10 


30  A 

Anode  Volts 

FIG.  118. 


50 


60 


Maxwell's  results  can  be  expressed  as  follows:  Let  F  and  P 
(Fig.  119)  be  two  infinitely  large  parallel  planes  with  a  grating  G 
of  parallel  wires  interposed  between  them.  Let  the  potentials 
of  F,  G  and  P  be  F/,  Vg  and  VPJ  respectively.  The  wires  of  the 
grid  are  of  the  same  thickness  and  have  a  radius  r,  the  distance 
between  the  wires  being  a.  Let/  and  p  be  the  distances  of  F  and  P 
from  the  grid.  The  assumption  is  made  that  /  and  p  are  large 

1 J.  C.  MAXWELL,  "  Electricity  and  Magnetism,"  Vol.  1,  p.  310. 

2  MAX  ABRAHAM,  Archiv.  fur  Elektrotechnik,  Vol.  8,  p.  42,  1919.  R.  W. 
KING,  paper  read  at  October  meeting  of  Am.  Phys.  Soc.,  Philadelphia,  1919. 
SOHOTTKY,  Archiv.  fur  Elektrotechnik,  Vol.  8,  p.  12,  1919.  v.  LAUE,  Ann  d. 
Phys.,  Vol.  59,  p.  465,  1919. 


228 


THERMIONIC  VACUUM  TUBE 


compared  with  a,  which  again  is  large  compared  with  r.  If  cy 
and  cr'  are  the  charge  densities  induced  on  F  and  P,  respectively, 
then 

r-V.-2-y,  .  (69) 


-i-V,,       .     (70) 
(71) 


Go 


where 


The  quantity  cr  gives  a  measure  of 
the  intensity  of  the  field  near  F,  which 
affects  the  flow  of  electrons  from  F. 
These  equations  apply,  of  course,  to  the 
case  in  which  there  are  no  electrons 
in  the  space  between  the  electrodes  to 
cause  a  distortion  of  the  field.  As  far 
as  the  determination  of  the  screening 
effect  of  the  grid  for  most  practical  pur- 
poses is  concerned,  it  is  found  that 
the  presence  of  the  electrons  in  the  space 
usually  does  not  materially  influence 

the  results.     From  these  equations  we  can  obtain  the  following 

interesting  results: 

Suppose,  first,  that  the  grid  wires  are  infinitely  thin,  so  that 

~  is  zero.     Then  a  is  infinitely  large  and  equation  (69)  reduces 

to  the  simple  form  applicable  to  two  plates  without  the  grid, 
namely, 

V,-VP.  (72) 


O 

FIG.  119. 


If,  on  the  other  hand,  the  grid  is  of  finite  dimensions,  but  G 
is  connected  to  F,  i.e.,  Vg=Vf,  then 


(73) 


From  these  two  equations  it  follows  that  the  charge  induced  on 
F,  when  the  grid  is  interposed  and  connected  to  F,  is  to  the 


THE  THERMIONIC  AMPLIFIER  229 

charge  induced  on  F  by  the  same  potential  on  P  when  there  is 
no  grid,  as  1  is  to 

1+  * 


This  result  expresses  the  screening  effect  of  the  grid  in  its 
simplest  form. 

Another  case  that  is  of  interest  is  the  relation  between  the 
stray  field  acting  through  the  grid  when  the  latter  is  connected 
to  F  and  the  field  obtained  when  grid  and  plate  P  are  connected 
together,  the  potential  applied  to  the  plate  P  being  the  same. 
This  is  readily  obtained  by  putting  V9=Va  in  equation  (69). 
Thus: 


V-  V,).    .    .    .     (74) 
Comparison  with  equation  (73)  gives 

^-3=l+2  ......  (75) 

(72  a 

What  we  are  interested  in  when  using  the  tube  in  an  a-c.  circuit 
is  the  relation  between  the  variation  in  the  field  produced  at  F 
by  a  variation  of  the  grid  potential  to  that  produced  by  an  equal 
variation  in  the  potential  of  the  plate,  both  fields  acting  at  the 
same  time.  This  can  be  obtained  directly  as  King  1  has  done, 
by  evaluating  the  partial  derivatives  of  <r  with  respect  to  Va 
and  Vp  (equation  69): 


Let  F'be  the  filament  of  a  vacuum  tube.  Then,  applying  the 
usual  notation,  Vf—  VP  =  EP  and  Vf—Vg=Eg,  and  substituting 
the  value  of  a  from  (71),  we  obtain  directly  from  equation  (76): 


dJijp  __     _ -.,  f, />77\ 

aloge  (2  sin—  J 


1  Loc.  cit. 


230 


THERMIONIC  VACUUM  TUBE 


When  the  diameter  of  the  wire  is  small  compared  with  the  distance 
between  adjacent  wires,  --is  small,  aud  we  can  write  approxi- 
mately: 


60 


50 


o 
o 

c 
o 


10 


where 


(78) 


a  log 


p  -.456  cm. 
r  -  .01  cm. 


L 


4  6 

H umber  of  Wires  Per  Cm. 

FIG   120. 


to 


p= distance  between  grid  and  plate; 

a  =  distance  between  adjacent  grid  wires; 

r  =  radius  of  grid  wire. 

This  equation  does  not  give  as  good  results  as  the  empirical 
equation  that  will  be  given  below  (equation  79).     But  for  values 


THE  THERMIONIC  AMPLIFIER 


231 


of  ju  ranging  from  about  2  to  20,  equation  (78)  can  be  used  for 
designing  tubes  with  a  sufficiently  high  degree  of  accuracy  for  most 
practical  purposes.  The  extent  of  the  agreement  between  cal- 
culated and  observed  values  is  shown  in  Figs.  120  and  121.  The 
points  indicate  observed  values,  while  the  smooth  lines  represent 
equation  (78).  Each  point  represents  the  average  of  a  number 
of  tubes.  The  deviation  at  the  higher  values  of  ju  where  the  wires 
are  close  together,  is  inherent  in  the  equation  which  was  derived 
on  the  assumption  that  the  distance  between  successive  wires 
is  large  compared  to  the  thickness  of  the  wires. 


HV 

£. 

1 
3 

la 

n=7.88 
r~  .01 

per  cm. 
cm. 

/ 

0 

jf 

s6 

/ 

6   L° 
o 

£ 
~C_ 

c  10 

0 

/ 

^ 

/ 

/* 

—  - 

) 

.Z                .3                .4                .5 

Distance  Between  Plate  and  Grfo^Cm. 
FIG.  121. 

On  account  of  the  accuracy  with  which  tubes  must  be  designed 
for  telephone  repeater  purposes,  the  author  carried  out  an  exten- 
sive series  of  measurements  in  1914,  to  establish  an  empirical  for- 
mula relating  the  tube  constants  with  its  structural  parameters. 
The  equation  which  was  found  to  give  the  best  results  is: 

M  =  Cprn2+l, (79) 

where 

p  =  distance  between  plate  and  grid; 

r— diameter  of  grid  wires; 

n  =  number  of  wires  per  unit  length. 


232 


THERMIONIC  VACUUM  TUBE 


C  is  a  constant  which  for  the  parallel-plane  type  of  tube  (see  Fig. 
68)  has  a  value  of  80.  Since  this  equation  is  non-dimensional, 
C  is  independent  of  the  system  of  units  used  in  expressing  the  tube 
dimensions  and  is  independent  of  the  size  of  the  tube  structure. 
It  will  be  recognized  that  this  equation  is  the  same  as  that 

given  on  page  44  where  d  =  p  and  k  =  ~ — ^ 


70 


6C 


Z 


p  = .  4  76  cm. 


1 


50 


40 


PP.. 


3 17cm. 


. 238  cm. 


./p  -  •  158cm. 


4  6  8  10  12  14 

Number  of  Grid  Wires  Per  Cm. 

FIG.  122. 

Equation  (79)  has  been  determined  from  measurements  made 
on  a  large  number  of  carefully  constructed  tubes  in  which  not  only 
the  quantities  given  in  the  equation  were  varied,  but  also  the 
istance  between  filament  and  plate,  and  the  distance  between 
Slament  and  grid  varied  over  a  wide  range.  The  constant  is, 
however,  independent  of  these  latter  two  distances,  as  the  equa- 
tion shows.  This  is  in  accordance  with  Maxwell's  result  which 
also  states  that  the  stray  field  between  filament  and  grid  is  inde- 


THE  THERMIONIC  AMPLIFIER 


233 


pendent  of  the  distance  between  them.  (See  Equation  75.)  Equa- 
tion (79)  is  more  accurate  than  the  theoretical  equation  (78)  and 
therefore  has  been  used  by  the  Western  Electric  Company  for  the 
design  of  substantially  all  its  tubes.  The  accuracy  with  which  this 
equation  holds  is  shown  in  Fig.  122,  where  /x— 1  is  plotted  as  a 
function  of  n,  the  number  of  wires  per  centimeter  length  of  the  grid 
for  various  distances  p  between  grid  and  plate.  The  curves  are 
computed  from  equation  (79),  while  the  circles  and  crosses  repre- 
sent the  observed  values  of  /*—  1. 

The  radius  r  of  the  grid  wires  in  these  tubes  was  1.02X10"2 
cm.  The  relation  between  /x  and  r  is  shown  in  the  following 
table,  which  also  contains  values  to  indicate  the  range  over  which 
distances  of  the  grid  and  plate  from  the  filament  were  varied. 
The  agreement  between  observed  and  computed  values  is,  as  will 

be  seen  from  the   table,   quite  good  for  values  of  -  ranging  to 

about  0.3.  The  thickest  grid  wire  used  in  these  tubes  had  a  radius 
of  2.54X10"2  cm.  It  is  usually  desirable  to  use  thin  wires,  unless 


Amplification  Constant  /*. 

p+f 

/ 

P 

n 

r 

Observed. 

Calculated. 

(eq.  79) 

.635 

.158 

.475 

5.12 

.0102 

10.8.. 

11.1 

.635 

.158 

.475 

5.12 

.0191 

18.0 

19.0 

.635 

.158 

.475 

5.12 

.0254 

25.8 

26.1 

.318 

.158 

.158 

8.26 

.0102 

8.4 

9.8 

.397 

.158 

.238 

8.26 

.0102 

14.7 

14.5 

.476 

.158 

.317 

8.26 

.0102 

20.2 

18.7 

.556 

.158 

.397 

8.26 

.0102 

23.2 

23.0 

.635 

.158 

.476 

9.84 

.0102 

42.0 

38.5 

.635 

.158 

.476 

6.7 

.0102 

18.1 

18.3 

.635 

.238 

.397 

8.26 

.0102 

24.6 

23.1 

.635 

.317 

.317 

8.26 

.0102 

16.5 

18.6 

.635 

.397 

.238 

8.26 

.0102 

14.0 

14.5 

.635 

.476 

.158 

8.26 

.0102 

11.0 

9.8 

.635 

.158 

.476 

11.4 

.0102 

50.5 

51.5 

p  =  distance  grid  and  plate  in  centimeters; 
/  =  distance  grid  and  filament  in  centimeters; 
n= number  of  wires  per  centimeter; 
r= radius  of  grid  wire  in  centimeters. 


234  THERMIONIC  VACUUM  TUBE 

requirements  of  rigidity  necessitate  the  use  of  heavy  wires,  such  as 
is  the  case  when  the  grid  is  in  the  form  of  a  helix,  supported  only 
at  the  ends,  or  sometimes  even  at  one  end  only. 

King  in  the  paper  referred  to  above  has  also  given  an  equation 
for  M  for  the  three  classes  of  cylindrical  structures  shown  in  Fig. 
123.  His  argument  applies  particularly  to  the  case  in  which  the 
grid  wires  are  parallel  to  the  axis  of  the  structure,  but  the  resulting 


p* 


o 
o 


equation  applies  almost  equally  well  when  the  grid  is  a  helix. 
The  equation  is: 


(80) 


2-jrnr 


where  n  —  number  of  grid  wires  per  unit  length; 

r  =  radius  of  grid  wires; 
PP,  Po  =  radii  of  anode  and  grid. 

As  in  the  case  of  parallel-plane  structures,  /x  does  not  depend 
on  the  distance  between  filament  and  grid.  The  negative  sign 
in  equation  (80)  is  to  be  used  for  the  type  of  tube  shown  in  Fig. 
123C. 

Equation  (80)  gives  a  reasonably  good  agreement  with  observed 
values  of  /i.  With  the  help  of  the  equations  given  above,  it  is 
possible  to  determine  beforehand  the  tube  dimensions  required 
to  give  the  desired  value  of  JJL. 

76.  Calculation  of  Plate  Resistance.  The  plate  resistance 
can  in  general  not  be  determined  with  such  simple  equations  as 
those  which  make  possible  the  calculation  of  ju.  But  in  designing 

tubes  it  is  necessary  to  make  the  ratio  — ,  where  rp  is  the  plate 

rp 

resistance,  as  large  as  possible.     For  any  given  value  of  M  it  is 
therefore  desirable  to  make  rp  as  small  as  possible. 


THE  THERMIONIC  AMPLIFIER 


235 


o 


Now  rp  can  be  decreased  by  decreasing  the  distance  p+f 
between  filament  and  plate.  On  the  other  hand  fj,  increases  with 
increasing  distance  p  between  grid  and  plate,  but  is  independent 
of  the  distance  /  between  filament  and  grid.  Hence,  in  order  to 
keep  M  large  and  rp  small,  /  should  be  kept  as  small  as  possible, 
i.e.,  the  grid  should  be  close  to  the  filament. 

The  plate  resistance  depends,  furthermore,  on  the  size  of  the 
electrodes.  It  is  within  certain  limits  inversely  proportional  to 
the  area  of  the  anode  as  well  as  that  of  the  cathode.  It  will  be 
evident  that  there  are  limitations  to  increasing  the  area  of  the 
anode.  For  example,  if  the  cathode  is  a  single  straight  filament 
and  the  anode  a  plane  parallel  to  the  filament, 
there  would  be  a  limit  to  the  size  of  the  anode 
beyond  which  any  further  increase  in  its  size  would 
not  contribute  appreciably  to  a  reduction  in  the 
resistance.  On  the  other  hand,  the  resistance  can 
be  reduced  very  much  by  using  two  plates,  one  on 
either  side  of  the  filament,  as  is  mostly  done.  If 
the  cathode  consists  of  more  strands  of  filament, 
the  anode  area  can,  of  course,  be  further  increased 
to  advantage. 

If  the  anode  is  cylindrical,  an  increase  in  its 
diameter  would  increase  the  distance  between  fila- 
ment and  anode  in  the  same  proportion  as  the 
anode  area  is  increased.  Considering  a  surface 
element  of  the  anode,  the  resistance  is  proportional 
to  the  square  of  the  distance  between  the  cathode 
and  the  anode  element.  But  for  a  cylindrical  anode 
the  area  can  be  increased  only  by  increasing  the 
radius  in  the  same  proportion,  so  that  the  resistance  increases 
linearly  with  the  area  of  the  anode;  or,  what  is  the  same  thing,  it 
increases  linearly  with  the  radius. 

In  this  connection,  we  may  note  an  interesting  relation  between 
cylindrical  and  parallel-plate  tubes,  which  was  pointed  out  by 
R.  W.  King.  Suppose  a  cylindrical  structure,  having  a  thin 
filament  stretched  along  the  axis  of  the  anode  (Fig.  124)  be  unfurled 
so  that  the  anode  becomes  a  plate  having  a  width  equal  to  2irpv. 
Let  the  filament  be  replaced  by  a  surface  equal  to  the  anode  area 
and  at  a  distance  pp  from  it.  Assuming  the  cathode  to  be  an 
equi-potential  surface,  the  space  current  per  unit  area  of  the 


FIG.  124. 


236  THERMIONIC  VACUUM  TUBE 

parallel  plane  structure  is  given  by  equation   (9)   Chapter  IV. 
Hence  the  current  for  this  tube  is  given  by 


PP 

where  I  is  length  of  the  structure  (perpendicular  to  the  paper). 
That  is 

7=14.65X10-6—  W/*. 
PP 

This  is  the  same  equation  that  applies  to  the  cylindrical  tube 
(see  equation  (14),  page  60).  The  two  structures,  therefore,  give 
the  same  space  current. 

As  regards  the  effect  of  the  area  of  the  cathode,  there  are  also 
certain  limitations.  If  the  area  of  the  cathode  be  increased  by 
increasing  its  diameter,  the  resistance  will  not  be  reduced  propor- 
tionately because  of  the  density  of  the  space  charge  of  the  electrons 
in  the  neighborhood  of  the  filament.  The  total  saturation  current 
will,  of  course,  be  greater,  but  will  only  be  obtained  at  a  higher 
voltage.  The  better  way  to  increase  the  area  of  the  cathode  is  to 
increase  its  length.  However,  this  generally  means  an  increase 
in  the  voltage  drop  in  the  filament,  due  to  the  heating  current,  and 
this  in  itself  increases  the  plate  resistance,  due  to  the  limitation 
of  the  current  by  the  filament  voltage.  (See  Fig.  20,  page  61.) 

King  has  also  derived  equations  for  the  space  current  as  a 
function  of  the  structural  parameters  for  both  cylindrical  and 
parallel-plane  structures,  on  the  assumption  that  the  current  can 
be  taken  to  vary  as  the  f-power  of  the  effective  voltage.  Since 

K 

the  effective  voltage  —+Eg  is  generally  not  large  compared  with 

the  voltage  drop  in  the  filament,  the  limitation  of  space  current 
by  the  latter  must  be  taken  into  account,  in  which  case  the  current 
will  be  governed  by  equations  (18)  and  (19)  of  Chapter  IV. 
If  the  voltage  drop  in  the  filament  be  neglected  the  computed  cur- 
rent will  in  general  be  considerably  larger  than  the  observed  cur- 
rent. 

77.  Types  of  Thermionic  Amplifiers.  The  number  of  different 
types  of  thermionic  tubes  now  in  use  has  become  so  large  that  no 
attempt  will  be  made  to  describe  or  even  mention  all.  The  purpose 
in  describing  any  is  merely  to  give  the  reader  a  quantitative  idea 


THE  THERMIONIC  AMPLIFIER  237 

of  characteristics  of  tubes  used  in  practice.  At  the  outset  it  may 
be  stated  that  tubes  are  used  ranging  from  a  type  that  consumes  for 
its  operation  a  small  fraction  of  a  watt,  to  types  that  give  several 
thousand  watts'  output  in  the  form  of  alternating  current.  This 
widely  varying  range  is  occasioned  by  the  widely  differing  condi- 
tions to  be  satisfied,  depending  on  the  purpose  for  which  the  tube  is 
to  be  used.  If  the  tube  is  to  operate  as  an  amplifier  with  the  tele- 
phone receiver  connected  directly  in  its  output  circuit,  the  output 
power  necessary  need  not  be  more  than  a  very  small  fraction  of  a 
watt — one  millionth  of  a  watt  is  quite  sufficient  to  give  a  very  loud 
tone  in  most  well-constructed  receivers.  If,  on  the  other  hand, 
the  tube  is  to  be  used  as  a  telephone  repeater,  inserted  at  a  point 
on  the  telephone  line  about  midway  between  the  sending  and  re- 
ceiving stations,  the  tube  must  give  a  sufficient  amount  of  powrer 
to  give  clearly  audible  speech  in  the  receiver  after  the  telephone 
currents  have  been  attenuated  by  the  line  between  the  repeater 
and  the  receiver.  Then,  again,  if  the  tube  is  used  to  amplify 
modulated  high-frequency  oscillations,  for  example,  before  being 
impressed  on  an  antenna  for  radio  transmission,  it  must  obviously 
be  capable  of  giving  a  much  larger  output  power,  the  magnitude 
of  which  depends  upon  the  distance  over  which  transmission  is  to 
take  place,  and  can  range  all  the  way  up  to  several  kilowatts. 
When  the  necessary  power  is  too  large  to  be  handled  by  one  tube, 
a  number  of  tubes  can  be  used  in  parallel. 

In  designing  tubes  for  amplification  purposes  several  factors 
have  to  be  taken  into  consideration.  It  is,  for  example,  necessary 
to  consider  the  output  power  that  is  necessary.  To  obtain  best 
operation  the  plate  resistance  of  the  tube  should  be  made  equal 
to  the  impedance  into  which  the  tube  works.  If  this  is  not  possi- 
ble or  desirable  from  the  point  of  view  of  tube  construction, 
a  transformer  could  be  used  in  the  output  circuit  to  match  the 
tube  resistance  on  the  one  side  and  the  line  impedance  or  the  imped- 
ance of  the  recording  apparatus  on  the  other.  It  is  also  possible 
to  use  two  or  more  tubes  in  parallel,  thus  reducing  the  total  plate 
resistance.  Referring  to  equation  (31)  it  will  be  seen  that  the 
output  depends  also  upon  the  input  voltage  eg  and  the  amplifica- 
tion constant  fi.  The  input  voltage  must  be  kept  within  the  limits 
defined  by  equations  (24)  and  (25).  Furthermore,  n  and  the  plate 
resistance  must  be  so  chosen  that  the  amplification  has  the 
desired  value.  This  is  usually  as  large  as  possible. 


238 


THERMIONIC  VACUUM  TUBE 


There  are  thus  a  number  of  requirements  to  be  satisfied  and  they 
differ  with  different  operating  conditions.  The  following  tubes 
represent  a  few  standard  types.  The  tube  characteristics  are 
specified  sufficiently  fully  by  giving  the  value  of  /z,  the  filament 
constants,  the  relation  between  plate  current  and  plate  voltage, 

S.8 
il 

'5.4 
5.Z 

40*IOJ 


\ 


E-0 


g- 


1.3 


100 


120 


140 


cu  4-0  60  80 

Anode  Volfs-Ep 

FIG.  125. 

and  the  relation  between  plate  resistance  and  plate  voltage, 
over  a  range  of  operating  plate  voltages.    The  slope  —  of  the 

Tv 

plate  current-grid  potential  characteristic  can  then  be  obtained 
directly  from  the  known  values  of  /*  and  rv. 

The  tube  shown  in  Fig.  68,  page  146  represents  a  modern  type 


THE  THERMIONIC  AMPLIFIER 


239 


of  telephone  repeater  manufactured  by  the  Western  Electric 
Company  and  used  on  the  lines  of  the  Bell  Telephone  System. 
The  overall  length  of  the  tube  and  base  is  about  4  inches.  The 
plates  are  of  nickel  and  their  edges  are  turned  up  to  prevent  warp- 
ing due  to  the  high  temperature  to  which  they  rise  when  bom- 


10 
8 

6 
5 
4 
5 

2 
I 

ii.o 
|.i 

.5 

0 
0 

o 

/ 

1 

o 

/ 

i 

/ 

\ 

1  < 

> 

A 

A 

L 

9 

1 

f 

1 

/ 

3 
o 

.1 

i 

8 

f 

/ 

/ 

I 

f 

10                    £0          iO       40     50    60      80     100                  20C 

Anode  Volts 

FIG.  126. 

barded  by  the  electrons  during  the  process  of  evacuation.  It 
contains  an  oxide-coated  platinum  filament  operating  on  a  normal 
filament  current  1.3  amperes,  the  voltage  being  7  volts.  When 
operated  as  a  telephone  repeater  the  d-c.  plate  voltage  is  160 
volts,  and  the  grid  is  maintained  negative  with  respect  to  the 
filament  by  a  battery  of  9  volts.  The  characteristics  are  shown 


240 


THERMIONIC  VACUUM  TUBE 


in  Fig.  125.  The  plate  resistance  is  shown  here  and  in  the  follow- 
ing curves  as  a  function  of  the  plate  potential,  the  grid  potential 
being  zero.  To  obtain  the  plate  resistance  for  any  grid  potential 
other  than  zero,  all  that  is  necessary  is  to  add  pEa  to  the  plate 
potential  and  read  the  resistance  from  the  curve  at  the  value  of 
plate  potential  equal  to  the  value  so  obtained.  Thus,  since  ju 
is  about  5.6,  the  plate  resistance  at  a  plate  potential  £^=160, 
and  grid  potential  E0=—$  is  that  corresponding  to  an  abscissa 
of  160-9X5.6=100  volts,  namely,  5000 
ohms.  The  minimum  amplification  required 
of  this  tube  is  25  miles  of  standard  cable, 
which  corresponds  to  a  power  amplifica- 
tion of  230.  (See  table  on  page  219.) 
The  logarithmic  plot  of  this  tube's  charac- 
teristic is  shown  in  Fig.  126.  The  slope 
of  the  line  is  close  to  2,  indicating  a  parabolic 
relation  between  current  and  voltage  over 
the  operating  range. 

Fig.  127  shows  a  type  of  tube,  com- 
monly known  as  the  VT-1,  that  is  suitable 
for  use  either  as  detector  or  amplifier  and  is 
designed  to  operate  on  a  plate  voltage  of 
about  30  volts  when  delivering  power 
directly  to  a  telephone  receiver.  Its  operat- 


Length:  10  cms. 
are  shown  in  Fig. 


FIG.  127.— Western  Elec-  ing  filament  current  and  voltage  are  1.1 
trie  Receiving  Tube,  amperes  and  2.5  volts,  and  its  amplifica- 
tion constant  is  6.  Its  other  characteristics 
128.  The  logarithmic  plot  of  the  charac- 
teristic is  shown  in  Fig.  129.  The  slope  of  this  line  is  also 
close  to  2.  The  minimum  amplification  at  30  volts  on  the 
plate  is  24  miles  of  standard  cable.  This  tube  was  manu- 
factured by  the  Western  Electric  Company  for  use  as  an  aero- 
plane radio  receiver.  The  aeroplane  radio  transmitter  tube 
resembles  in  its  structural  features  the  one  shown  in  Fig.  68 
but  was  designed  to  operate  on  plate  voltages  ranging  from  275 
to  350  volts  instead  of  160  volts,  the  voltage  of  the  telephone 
repeater.  The  evident  rugged  construction  of  these  tubes  was 
found  necessary  to  enable  them  to  withstand  the  rather  severe 
vibration  to  which  they  are  subjected  on  an  aeroplane.  In  the  case 
of  the  receiver  tube  (Fig.  127)  the  filament,  plate  and  grid  are  sup- 


THE  THERMIONIC  AMPLIFIER 


241 


ported  from  the  top  by  means  of  a  block  of  lavite.  The  lower 
part  of  the  plate  forms  a  collar  which  fits  tightly  round  the  re- 
entrant tube.  The  double  plates  and  grids  are  each  stamped  in 
one  piece  from  sheet  metal,  thus  facilitating  quantity  production. 
A  tube  of  simple  construction  manufactured  by  the  General 
Electric  Company  is  shown  in  Fig.  130.  The  anode  consists  of  a 
nickel  cup  about  -f?  inch  in  diameter  and  -3%  inch  high,  and  is 


4.5 

4.0 
•>  ^ 

j 

I 

/ 

Fif.  Amps  =1.2 
Grid  Volts  =  0 

/ 

35,000 
30,000  V 

0 

1 

_ 
3.0 

_ 

2.5 

,, 

2.0 
1.5 
1.0 
6.5 
Q 

, 

' 

I 

,<y 
v 

\ 

J 

1 

' 

o: 

• 

?oooo  c 

\ 

f 

15,000 

10000 

£ 

/ 

•^^ 

-o—^ 
1 

_t 

V 

j;^  J0_  20  30  40  50  60 

Anode  Volts -E£ 

FIG.  128. 


placed  over  the  grid  and  filament.  The  grid  is  in  the  form  of  a 
helix  enclosing  the  filament  which  is  likewise  helical  and  consists 
of  tungsten  wire. 

In  one  type  of  tube  of  this  construction  the  normal  operating 
filament  current  is  1.1  amperes  and  the  filament  voltage  3.6  volts. 
The  amplification  produced  by  the  tube  is  equivalent  to  about 
20  miles  of  standard  cable. 

Tubes  that  are  used  as  amplifiers  and  radio  detectors  and  in- 


242 


THERMIONIC  VACUUM  TUBE 


tended  to  deliver  power  directly  to  a  telephone  receiver  need  not 
be  capable  of  handling  more  power  than  is  necessary  to  operate 
the  receiver  and  in  most  cases  this  corresponds  to  a  very  small 
plate  current.  This  makes  it  unnecessary  and  undesirable  to 


•j.v 
4.0 

3.0 
2.0 

1.0 
.8 

.5 
.4 

.3 
2 

( 

) 

L 

O 

7' 

1° 

Fil.  Amps.  - 
Grid  Volts* 

0 

/ 

7 

1 

/ 

C 

/ 

/ 

.0                      10            30        40            60       80 
Anode  Volts 

FIG.  129. 

dissipate  more  than  a  fraction  of  a  watt  in  heating  the  filament. 
The  desirability  of  reducing  the  power  necessary  for  heating  the 
filament  is  apparent.  A  tube  in  which  this  has  been  done  is  shown 
in  Fig.  131.  The  filament  is  of  the  oxide-coated  type,  2X10~3 
inches  in  diameter  and  is  designed  to  operate  on  one  dry  cell,  the 


THE  THERMIONIC  AMPLIFIER 


243 


operating  voltage  ranging  from  1.0  to  1.5  amperes  with  a  normal 
filament  current  of  about  0.2  ampere.  The  normal  power  dis- 
sipated in  the  filament  is  therefore  about  one-quarter  of  a  watt 
This  tube  is  a  good  detector  and  gives  about  20  miles  amplification 
with  a  plate  voltage  of  about  30  volts.  It  was  designed  not  only 
for  the  purpose  of  reducing  as  __ 

much  as  possible  the  power  con- 
sumption in  the  filament,  but 
also  to  facilitate  the  construc- 
tion. The  four  stout  wires 
which  connect  to  the  electrodes, 
are  moulded  in  a  glass  bead  11, 
or  other  insulating  substance. 
This  forms  a  unit  on  which  the 
rest  of  the  structure  is  built. 
The  grid  is  sufficiently  rigid  to 
be  supported  only  from  the 
lower  end.  The  plate  is  likewise 
supported  by  a  single  wire  going 
through  the  glass  bead.  This 
means  of  supporting  gives  a 
sufficiently  rugged  structure  on 
account  of  its  small  size  and 
light  weight.  The  structure 
thus  mounted  on  the  bead  11 
is  inserted  into  a  tube,  the 
lower  end  of  which  is  pressed, 
as  shown  at  2,  Fig.  131,  thus 
making  an  air-tight  seal.  This 
method  of  constructing  the  tube 
eliminates  the  making  of  flares  and  presses. 

Another  type  of  tube  of  simple  construction  is  shown  in 
Fig.  132.  This  drawing  was  made  from  one  of  several  tubes 
designed  and  used  by  the  author  in  his  work  on  vacuum  tubes, 
about  six  years  ago.  The  same  type  was  subsequently  developed 
in  Europe  and  used  extensively  in  military  operations  during  the 
war.  The  simplicity  of  construction  of  this  tube  lies  in  the  hori- 
zontal mounting  of  the  electrodes. 

Turning  now  to  a  consideration  of  the  high-power  type  of 
tube,  it  must  be  noted  that  this  type  of  tube  belongs  to  a  class 


FIG.    130. — General    Electric    type 
Receiving  Tube.  Length:  10  cms. 


244 


THERMIONIC  VACUUM  TUBE 


that  is  entirely  different  from  the  small  types  discussed  above, 
and  presents  problems  that  make  its  construction  more  difficult. 
For  example,  on  account  of  the  large  amount  of  power  dissipated 
in  such  a  tube,  the  electrodes  and  walls  of  the  vessel  must  be  more 
thoroughly  freed  of  gas  during  evacuation  than  is  required  of  the 
low-power  type  of  tube.  The  heating  of  the  plates  during  operation 

makes  it  necessary  to  compute  their 
size  on  the  basis  of  the  power  dissi- 
pated at  them  by  electron  bom  bard- 
ment.  This  can  be  done  in  the 
manner  explained  in  Chapter  IV.  On 


FIG.  131. 


FIG.  132. 


account  of  their  low  vapor  pressure,  the  refractory  metals  such  as 
tungsten  and  molybdenum  are  very  suitable  materials  for  use  in 
high  -power  tubes.  Tungsten  can,  for  example,  be  heated  during 
evacuation  to  about  2500°  K.  to  drive  out  occluded  gases,  while 
during  operation  the  tungsten  anodes  can  safely  rise  to  a  tempera- 
ture of  about  1500°  K.,  thus  making  it  possible  to  use  a  smaller 
anode  area  than  is  necessary  when  less  refractory  metals  are  used  as 
anodes.  On  the  other  hand,  high-power  tubes  that  are  to  operate 


THE  THERMIONIC  AMPLIFIER 


245 


•«»^j 

3% 


on  comparatively  low  plate  voltages  often  require  so  much  fila- 
ment that  when  the  filament  strands  are  properly  spaced  and 
supported  in  a  mechanically  convenient 
manner,  they  occupy  a  surface  wrhich 
is  so  large  as  to  necessitate  a  large 
plate  area.  It  is  also  necessary,  in  de- 
signing power  tubes,  to  consider  the 
size  of  the  bulb  and  make  it  large  enough 
to  prevent  overheating  of  the  glass. 
The  radiating  area  of  the  glass  bulb, 
expressed  in  square  inches,  should  be  at 
least  equal  to  the  number  of  watts  dis- 
sipated inside  the  bulb. 

Fig.  133  shows  a  type  of  power  tube 
that  was  used  in  1915  by  the  American 
Telephone  &  Telegraph  Company  and 
the  Western  Electric  Company  to  trans- 
mit speech  from  Arlington,  Va.,  U.  S.  A., 
to  Paris  and  to  Honolulu,  a  distance  in 
the  latter  case  of  about  5000  miles. 
The  anode,  it  will  be  observed,  consists 
of  metallic  ribbon  so  arranged  on  glass 
frames  that  it  can  be  heated  during 
evacuation  by  passing  a  current  through 
it.  This  is  of  considerable  help  in  driv- 
ing out  occluded  gases.  A  glass  frame 
situated  midway  between  the  frames  on 
which  the  anodes  are  mounted  serves  to 
support  the  grid  and  filament. 

A  General  Electric  Company  type  of 
tube  which  also  enables  the  anode  to  be 
heated  by  passing  a  current  through  it  is 
shown  in  Fig.  134. l  Here  the  anode  con- 
sists of  tungsten  wire  stretched  back  and 
forth  on  a  glass  frame.  The  grid  consists 
of  fine  tungsten  wire  wrapped  around 
another  glass  frame. 

A  type  of  power  tube  of  British  design  is  shown  in  Fig.  135. 
The  electrodes   are  supported,   as  indicated,   by  springs  fitting 
1 1.  LANGMUIB,  General  Electric  Review,  Vol.  18,  p.  335,  1915. 


FIG.  133.— Western  Elec- 
tric Type  Power  Tube, 
used  in  early  Long- 
distance Radio-phone 
Experiments.  Length : 
30  cms. 


246 


THERMIONIC  VACUUM  TUBE 


tightly  in  the  side  elongations  of  the  bulb.  This  type  of  tube  ope- 
rates on  a  plate  voltage  of  several  thousand  volts. 

A  high-voltage  tube  of  more  rugged  construction  is  the  General 
Electric  type  shown  in  Fig.  136.  The  anodes  are  stout  tungsten 
plates,  about  2X2^  inches,  and  the  grid  consists  of  fine  tungsten 
wire  wrapped  around  a  molybdenum  frame,  thus  encasing  the 
filament.  This  tube  operates  normally  on  about  1500-2000  volts, 
and  is  capable  of  delivering  several  hundred  watts  a-c.  output, 
the  space  current  ranging  from  150  to  250  milliamperes. 

The  power  obtainable  from  a  tube  can  be  greatly  increased 
by  designing  the  tube  to  operate  on  high  voltages.  This,  however, 
requires  ordinarily  a  d-c.  source  of  high  voltage.  This  can  be 


obtained  either  by  using  a  valve  rectifying  system,  as  explained  in 
Chapter  VI,  or  by  using  d-c.  generators.  Such  sources  of  high  volt- 
age, direct  current  are  usually  not  efficient.  But  the  power  obtain- 
able from  a  tube  can,  of  course,  also  be  increased  by  increasing 
the  space  current;  that  is,  by  increasing  the  area  of  the  cathode 
or  its  thermionic  efficiency.  (See  page  78.)  The  coated  type  of 
cathode  has  a  high  thermionic  efficiency  and  therefore  lends  itself 
well  to  the  construction  of  high-power,  low-voltage  tubes.  'Such 
a  tube  is  shown  in  Fig.  137.  This  tube  is  capable  of  giving  an 
a-c.  output  of  several  hundred  watts  when  operating  on  a  plate 
voltage  of  only  about  700-1000  volts. 

In  order  to  avoid  using  large  containers,  we  can  use  cooling 
means  to  carry  off  the  heat  dissipated  at  the  anode,  instead  of  de- 
pending only  on  radiation.  Cooling  can  be  effected  in  several 


THE  THERMIONIC  AMPLIFIER 


247 


ways:  The  device  can,  for  example,  be  immersed  in  a  liquid  bath 
or  a  blast  of  air  can  be  d  rected  against  it.  Or  the  anode  can  be 
so  constructed  that  it  can  be  cooled  by  water  circulation.  A 
Western  Electric  type  of  tube  in  which  this  is  done  is  shown  in 
Fig.  138.  The  anode  consists  of  a  metal  tube  about  half  an  inch 


J 


FIG.  135.— British  Type  Power  Tube. 

in  diameter.  The  bottom  of  this  tube  is  closed  and  the  top  sealed 
hermetically  to  the  re-entrant  glass  tube,  thus  allowing  water  to  be 
circulated  inside  of  it.  The  grid  forms  a  helix  round  the  anode, 
and  the  cathode  consists  of  a  number  of  filaments  arranged  outside 
the  grid  to  form  a  cylindrical  system  concentric  with  the  grid 
and  anode.  Tubes  of  this  type  have  been  made  to  dissipate 
several  kilowatts. 


248 


THERMIONIC  VACUUM  TUBE 


Other  means  have  been  used  for  preventing  the  power  dis- 
sipation at  the  anode  from  raising  its  temperature  too  high.  One 
way  is  to  make  the  area  of  the  anode  as  large  as  possible,  but  to 
this  there  are  limitations.  For  example,  increasing  the  area  of 


L  .-' 


FIG.  136. — General  Electric  Type 
Power  Tube. 


FIG.  137. — Western  Electric  Type 
Power  Tube.     Length:  30  cms. 


a  cylindrical  anode  by  increasing  its  diameter  causes  an  increase 
in  the  resistance  of  the  tube.  This  is  avoided  in  a  type  of  tube 
of  General  Electric  design.  The  anode  is  shaped  in  the  form  of 
a  cylinder  having  four  radial  flanges,  which  help  to  increase  the 


radiating  area. 


THE  THERMIONIC  AMPLIFIER 


249 


An  obvious  way  of  keeping  the  temperature  of  the  anode  low 
is  to  increase  its  thermal  ernissivity  by  blackening  its  surface. 
(Equation  23,  Chapter  IV.) 

78.  Amplification  Circuits.  A  great  variety  of  circuit  arrange- 
ments have  been  devised  in  which  the  thermionic  tube  can  be 
operated  as  an  amplifier.  We  shall,  however,  discuss  only  a  few 


FIG.  138. — Power  Tube  with  Water-cooled  Anode. 

types  of  circuits  for  the  purpose  of  illustrating  the  points  that 
must  be  considered  in  designing  efficient  circuits.  The  necessary 
considerations  follow  directly  from  the  theoretical  discussions  in 
the  foregoing.  For  example,  it  was  shown  that  the  current  in  the 
circuit  can  be  controlled  by  potential  variations  impressed  on  the 
grid.  It  is,  therefore,  necessary  to  make  the  potential  varia- 
tions impressed  on  the  grid  as  large  as  possible.  For  this 


250 


THERMIONIC  VACUUM  TUBE 


purpose  the  step-up  transformer  T\  (Fig.  139)  is  inserted  in  the 
input  circuit.  The  secondary  of  this  transformer  should  be  wound 
to  have  the  highest  possible  impedance.  The  input  transformers 
that  are  used  in  connection  with  telephone  repeater  tubes  on  the 
Bell  telephone  lines,  have  a  secondary  impedance  of  600,000  ohms 
at  a  frequency  of  800  cycles  per  second.  It  is  desirable  to  shunt 
the  secondary  of  the  input  transformer  with  a  resistance  ra  about 
equal  to  the  secondary  impedance.  The  output  transformer  T<z 
is  so  wound  that  the  tube  works  into  an  impedance  equal  to  its 
plate  resistance,  the  secondary  impedance  of  To  being  equal  to  the 
impedance  of  the  line  or  device  to  which  energy  is  delivered.  If 
the  line  impedance  is  equal  to  the  plate  resistance  of  the  tube,  this 
transformer  can  be  omitted.  The  choke  coil  L  and  condenser  C 


FIG.  139. 

are  inserted  if  the  direct  'current  established  in  the  plate  circuit 
by  the  battery  Eb  is  too  large  to  pass  directly  through  the  winding 
of  the  output  transformer  without  causing  undue  heating  of  its 
coils.  The  grid  is  maintained  at  a  negative  potential,  with  respect 
to  the  filament,  by  the  battery  Ea,  which  should  be  so  adjusted 
with  respect  to  the  plate  voltage,  the  constants  of  the  tube,  and 
the  peak  value  of  the  input  voltage  that  the  limit  equations 
(24)  and  (25)  are  satisfied. 

Instead  of  using  a  separate  grid  battery,  the  grid  can  be 
maintained  negative  with  respect  to  the  filament  by  making  use 
of  the  voltage  drop  in  the  filament  rheostat.  The  arrangement 
is  shown  in  Fig.  140.  In  this  case  the  plate  circuit  is  connected 
to  the  positive  terminal  of  the  filament  battery,  thus  increasing 
the  potential  difference  between  the  filament  and  plate  by  an 
amount  equal  to  the  voltage  of  the  filament  battery.  This  is 


THE  THERMIONIC  AMPLIFIER 


251 


sometimes  desirable  since  the  amplification  increases  with  the 
plate  voltage. 

It  was  shown  in  Chapter  III,  that  there  exists  an  intrinsic 
potential  difference  between  the  filament  and  grid,  the  value  of 
which  depends  largely  upon  the  nature  of  the  surfaces  of  these 
electrodes.  This  quantity  was  designated  in  the  equations  by 
e.  Now,  e  can  be  either  positive  or  negative,  and  is  seldom 
greater  than  1  volt.  If  it  is  negative,  the  grid  will  be  intrinsically 
at  a  negative  potential  with  respect  to  the  filament.  If  this  is 
the  case  and  the  alternating  input  voltage  is  small  no  extra  pro- 
vision need  be  made  to  keep  the  grid  negative.  If  €  is  very  small 
or  has  a  positive  value,  the  grid  battery  should  be  inserted  or  a 
scheme  can  be  resorted  to  which  is  illustrated  in  Fig.  141  and  for 


FIG.  140. 


FIG.  141. 


which  I  am  indebted  to  my  associate  Mr.  R.  H.  Wilson.  The 
scheme  consists  essentially  in  the  addition  of  a  resistance  r  in  the 
d-c.  branch  of  the  output  circuit.  A  positive  value  of  e  causes 
electrons  to  be  attracted  to  the  grid  and  this  results  in  a  decrease 
in  amplification  due  to  a  reduction  in  the  effective  alternating 
grid  voltage,  as  explained  on  page  167.  But  a  positive  value  of  e 
also  increases  the  direct-plate  current.  This  increases  the  voltage 
drop  in  the  resistance  r  which  consequently  makes  the  grid  more 
negative  with  respect  to  the  filament.  The  more  negative  e  is, 
the  smaller  will  be  the  space  current  flowing  in  r  and  this  tends  to 
decrease  the  negative  potential  on  the  grid.  In  this  way  the 
circuit  tends  to  effect  a  balance,  so  that  tubes  having  different 
values  of  e  will  not,  when  used  in  this  circuit,  give  widely  varying 
degrees  of  amplification.  It  will  be  recognized  that  there  is  an 
optimum  value  for  the  resistance  r,  because  if  r  be  made  too  large 


252 


THERMIONIC  VACUUM  TUBE 


the  voltage  drop  in  it  would  make  the  grid  so  much  negative  with 
respect  to  the  filament  and  consequently  the  impedance  of  the 
tube  so  high  that  the  amplification  drops. 

The  effect  of  the  resistance  is  illustrated  in  the  following  table, 
which  gives  the  amplification  in  miles  of  standard  cable  obtained 
from  a  number  of  tubes  having  different  values  of  e.  The  best 
value  of  resistance  according  to  this  table  is  about  1000  ohms. 


Amplification  in  Miles  of  Standard  Cable. 

r  =  0 

r=400 

r  =  800 

r  =  1200 

r  =  2000 

r=4000 

+0.16 

13 

22 

24 

24 

25 

24 

+0.11 

14 

20 

24 

25 

25 

24 

-0.13 

14 

22 

26 

26 

26 

24 

-0.85 

21 

21 

21 

21 

20 

19 

-0.69 

22 

21 

21 

21 

20 

18 

-0.27 

24 

24 

24 

23 

23 

22 

Tubes  are  frequently  used  in  cascade  formation  to  secure 
higher  amplification  in  the  so-called  multi-stage  amplifier  sets. 
There  are  numerous  circuit  arrangements  whereby  this  can  be 


FIG.  142. 


done.  Fig.  142  shows  a  two-stage  set  in  which  the  first  tube  is 
operated  as  a  voltage  amplifier.  The  inductance  L  has  an  imped- 
ance which  is  large  compared  with  the  plate  resistance  of  the  first 
tube,  thus  permitting  the  largest  possible  voltage  amplification. 
It  was  shown  in  Section  G2  that  if  Lo>  is  twice  as  large  as  the  plate 


THE  THERMIONIC  AMPLIFIER 


253 


resistance  of  the  tube  to  which  it  is  connected,  the  voltage  ampli- 
fication is  about  90  per  cent  of  its  maximum  value  p.  If,  however, 
L  is  a  practically  pure  reactance,  it  is  desirable  to  make  it  more 
than  twice  as  large  as  the  plate  resistance,  in  order  to  make  the 
phase  difference  between  grid  and  plate  potentials  as  near  as  180° 
as  possible,  thereby  straightening  out  the  tube  characteristic  and 
minimizing  distortion.  (See  Section  59.)  The  circuit  shown 
in  Fig.  142  is  so  arranged  that  both  tubes  can  be  operated  from 
the  same  plate  battery.  The  grids  can  be  maintained  negative 
with  respect  to  their  adjacent  filaments  by  means  of  grid  batteries 
(not  shown)  or  by  connecting  them  to  convenient  points  on  the 


FIG.  143. 

filament  rheostat  Rf.  The  resistance  r  should  be  large,  preferably 
of  the  order  of  one  or  two  megohms,  and  merely  serves  the  pur- 
pose of  maintaining  the  grid  of  the  second  tube  at  the  desired  d-c. 
potential. 

Instead  of  using  the  inductance  L  and  condenser  C,  a  step-up 
transformer  can  be  inserted  between  the  tubes,  as  shown  in 
Fig.  143.  When  this  is  done  both  tubes  should  be  operated  as 
power  amplifiers  (see  page  184).  The  primary  impedance  of  the 
inter-tube  transformer  should  therefore  be  equal  to  the  plate 
resistance  of  the  first  tube  and  its  secondary  should  be  wound  to 
impress  the  highest  possible  voltage  on  the  grid  of  the  second  tube. 

The  thermionic  tube  makes  it  possible  to  obtain  high  degrees 
of  amplification  with  non-inductive  circuits  by  means  of  an 
arrangement  suggested  by  H.  D.  Arnold.1  Fig.  144  shows  a 
1  U.  S.  Patent  1129943,  1915. 


254 


THERMIONIC  VACUUM  TUBE 


non-inductive  amplifier.  Instead  of  the  inductance  L  a  non- 
inductive  resistance  r  is  used.  If  the  grid  battery  Eg  were 
omitted  the  grid  of  tube  B  would  be  at  the  same  potential  as  the 
plate  of  tube  A.  The  grid  of  B  would  therefore  be  positive  with 
respect  to  its  filament  by  an  amount  equal  to  the  potential  differ- 
ence between  filament  and  plate  of  tube  A.  To  avoid  this  the 
negative  voltage  Eg  is  applied  to  the  second  grid  to  give  it  the 
appropriate  negative  potential  with  respect  to  its  filament. 

This  non-inductive  type  of  amplification  circuit  is  a  very 
important  contribution  made  possible  by  the  thermionic  tube, 
because  it  enables  us  to  produce  almost  any  degree  of  amplification 
without  the  use  of  transformers.  In  many  instances  transformers 
are  undesirable.  This  is,  for  example,  the  case  when  dealing  with 


FIG.  144. 

currents  of  very  low  frequency,  such  as  are  used  on  telegraph  lines 
and  especially  on  submarine  telegraph  cables.  Transformers  for 
such  frequencies  are  unpractical,  being  costly  and  inefficient. 
Besides,  they  distort  the  wave  form  which  it  is  very  desirable  to 
preserve.  Even  when  dealing  with  currents  of  frequencies  cov- 
ering the  audible  range,  transformers  produce  distortion  which  in 
some  cases  is  very  serious.  This  can,  for  example,  happen  in 
the  transmission  of  music.  When  speech  is  transmitted  through 
a  system  having  a  transmission  band  ranging  from  a  few  hundred 
to  about  2000  cycles  per  second,  the  speech  is  still  perfectly 
intelligible  and,  in  fact,  a  smaller  frequency  range  often  suf- 
fices. Transformers  that  are  used  on  telephone  lines  have  a  fairly 
flat  frequency  characteristic  and  are  very  satisfactory  for  speech 
transmission.  But  for  the  transmission  of  music  a  much  wider 
range  of  frequencies  is  necessary.  It  is  known  that  to  preserve 


THE  THERMIONIC  AMPLIFIER  255 

the  quality  of  many  musical  tones,  the  system  must  be  capable 
of  transmitting  with  equal  facility  frequencies  ranging  up  to 
several  thousand  cycles  per  second.  In  all  such  cases  the  non- 
inductive  amplification  circuit  is  of  value.  Care  must,  of  course, 
be  taken  to  eliminate  the  distortion  produced  by  the  curvature  of 
the  characteristic  of  the  tube  itself. 

In  ordinary  circuits,  such  as  that  shown  in  Fig.  139,  trans- 
formers and  coils  are  used  for  convenience  to  serve  definite  pur- 
poses. The  output  transformer  T%  is,  for  example,  inserted  to 
match  the  impedance  of  the  line  or  device  into  which  the  tube 
works  with  the  impedance  of  the  tube.  This  secures  maximum 
power  amplification.  The  input  transformer  likewise  matches  the 
impedances  and  has  a  very  high  secondary  impedance  because  of 
the  high  input  impedance  of  the  tube.  Referring  to  equation  (32) 
(page  187),  it  will  be  noticed  that  the  power  amplification  rj  is 
directly  proportional  to  the  input  resistance  rff.  Hence,  if  the  tube 
is  to  amplify  currents  from  a  low  impedance  line  and  the  input 
transformer  TI  were  omitted,  the  amplification  would  be  very 
small.  To  overcome  this  Arnold  suggested  using  voltage  ampli- 
fier tubes  to  step  up  the  input  voltage.  These  tubes  then  feed 
into  tubes  having  an  impedance  sufficiently  low  to  be  connected 
directly  to  the  output  circuit.  This  can  be  done  by  designing 
tubes  to  have  a  low  plate  resistance  or  using  a  number  of  tubes  in 
parallel. 

Telephone  transformers  that  are  commonly  used  on  the 
input  side  of  vacuum  tubes  have  voltage  step-up  ratios  ranging 
from  about  18  to  40.  Voltage  amplifier  tubes  having  an  ampli- 
fication constant  ;u=40  are  commonly  used.  It  follows  from 
equation  28,  page  183;  that  if  the  resistance  r  (Fig.  144)  is  made 
five  times  as  large  as  the  plate  resistance  rp  of  the  tube  A,  the 
voltage  amplification  produced  by  this  tube  is  33.  •  It  can,  there- 
fore, take  the  place  of  the  input  transformer.  The  voltage  can, 
of  course,  be  amplified  still  more  by  increasing  the  number  of 
tubes  in  the  cascade  series. 

Fig.  144  shows  the  filaments  connected  in  parallel  to  a  common 
battery.  They  can,  of  course,  also  be  operated  in  series  from  a 
common  battery.  Which  ever  is  the  more  desirable  depends  upon 
the  filament  battery  available.  Both  arrangements  are  used 
where  tubes  are  operated  in  parallel  to  give  increased  output. 
But  when  the  filaments  are  connected  in  series  and  the  grids  in 


256  THERMIONIC  VACUUM  TUBE 

parallel,  provision  must  be  made  to  counteract  the  difference  in 
potential  between  the  successive  grids  and  filaments,  due  to  the 
voltage  drop  in  the  preceding  filaments.  This  is  done  by  the  inser- 
tion of  appropriate  grid  batteries. 

In  designing  multi-stage  amplifier  circuits,  it  is  very  important 
to  make  sure  that  all  the  tubes  operate  in  accordance  with  the 
limit  equations  (24)  and  (25).  In  all  cases  the  voltages  impressed 
on  the  inputs  of  the  tubes  must  be  as  high  as  possible,  irrespective 
of  what  the  input  power  may  be,  because  the  power  developed 
in  the  output  depends  primarily  on  the  input  voltage.  Now,  when 
using  a  tube  as  a  voltage  amplifier  it  must  be  designed  to  have  a 
large  amplification  constant  /z.  This,  according  to  equation  (27) 
produces  a  large  voltage  amplification.  But,  referring  to  equation 
(25),  it  is  seen  that  the  larger  the  value  of  /x  the  smaller  is  the  input 
voltage  eg  that  can  for  constant  plate  battery  voltage  Eb  be 
impressed  on  the  input  without  causing  distortion.  When  it  is 
necessary  to  use  a  multi-stage  amplifier  set  the  input  voltage  on 

the  first  tube  is  generally  so  small  that  //  for  the  first  tube  can  be 

-pi 
quite  high  and  the  plate  voltage  not  large,  because  --  need  not 

be  larger  than  about  2e0,  where  Et,  is  the  voltage  of  the  plate 
battery.  But  the  voltage  impressed  on  the  second  tube  after 
being  amplified  by  the  first  tube  is  then  very  much  larger  and  the 
second  tube  must  be  capable  of  handling  this  increased  voltage. 
If  the  first  tube,  operating  on  a  definite  plate  battery,  is  just  capable 
of  handling  the  voltage  impressed  on  its  input,  then,  in  order  to 
handle  the  amplified  voltage,  the  second  tube  must  be  designed 
to  have  a  lower  /*  or  otherwise  must  operate  on  a  higher  plate 
voltage  than  the  first,  so  that  it  operates  on  a  characteristic  having 
a  larger  intercept  on  the  axis  of  grid  potential.  (See  Fig.  74, 
page  152.)  This  often  necessitates  heating  the  filament  of  the 
second  tube  to  a  higher  temperature  to  increase  the  range  of  the 
characteristic.  Such  considerations  show  the  important  part 
played  by  the  structural  parameters  of  the  tube.  In  some  multi- 
stage amplifier  sets  it  is  possible  to  use  several  like  tubes  in  series 
operating  on  the  same  plate  voltage;  that  is,  when  the  input  volt- 
age is  much  smaller  than  the  limiting  voltage  that  the  first  tube  can 
handle.  For  example,  if  the  amplification  constant  M  is  40,  and 
the  plate  voltage  is  120  volts,  the  intercept  of  the  IpEg-curve  is 
equivalent  to  3  volts.  If  E0=1.3  volts,  the  input  voltage  can 


THE  THERMIONIC  AMPLIFIER  257 

therefore  have  a  maximum  value  of  about  1.5  volts,  since  the  grid 
can  generally  be  allowed  to  become  slightly  positive.  If  this  is 
the  input  voltage  on  the  first  tube,  the  succeeding  tubes  must 
have  lower  amplification  constants  or  must  operate  with  higher 
plate  voltages.  But  if  the  input  voltage  on  the  first  tube  is,  say, 
5X10~5  volt,  it  can  be  amplified  30,000  times  before  it  becomes 
too  large  to  be  handled  by  this  type  of  tube  operating  under  the 
conditions  specified  above.  If  each  tube  with  /z  =  40  operates  with 
an  external  resistance  equal  to  four  times  its  plate  resistance, 
it  produces  a  voltage  amplification  of  32.  It  would  therefore  take 
three  such  tubes  in  series  to  produce  a  voltage  amplification  of 
30,000,  and  all  three  tubes  can  operate  under  the  above  conditions. 

It  may  be  remarked  that  if  a  is  the  voltage  amplification 
produced  by  one  tube,  the  total  amplification  A  produced  by  n 
tubes  is  A  =  an. 

Instead  of  using  a  large  number  of  tubes  in  cascade  to  produce 
a  high  degree  of  amplification,  use  could  be  made  of  a  " feed-back" 
arrangement,  due  to  R.  V.  L.  Hartley,1  which  is  of  special  advan- 
tage when  large  amplification  is  to  be  produced  with  a  few  tubes 
in  a  non-inductive  circuit.  It  will  be  evident  that  in  any  ampli- 
fying system  the  power  developed  in  the  output,  which  is  larger 
than  that  in  the  input,  can  be  greatly  increased  by  feeding  a  small 
portion  of  the  energy  in  the  output  back  to  the  input,  thus  ream- 
plifying  that  portion.  This  increases  the  output  power  and  also 
the  portion  fed  back  to  the  input,  and  in  this  way  the  original 
input  power  can  be  amplified  to  almost  any  desired  extent  depend- 
ing on  the  portion  fed  back  and  the  limits  of  the  tube  characteristic. 

Thus,  suppose  unit  power  be  applied  to  the  input  of  an  ampli- 
fying system,  the  normal  amplification  of  which  is  a-fold.  The 
power  in  the  output  is  then  a.  Let  a  fraction  s  of  the  output 
power  be  fed  back  to  the  input,  so  that  the  power  returning  to 
the  input  is  as.  The  portion  remaining  is  a  (1  —  s).  The  fraction 
as  amplified  again  into  the  output  becomes  a?s.  Of  this,  a  portion 
a?s  X  s  =  a?s2  is  again  fed  back  to  the  input,  leaving  a2s  —  a2s2  =  a?s 
(1—s)  available  in  the  output.  This  process  is  repeated  and  we 
get  in  the  output  an  amount  of  power  given  by  the  sum  of  a  series 
of  which  the  (n+l)th  term  is  a(l-s)ansn.  Thus,  if  A  is  the  total 
output  power: 

A=a(l-s)^l+as+a2s2+  .  .  .  aV  .  .  .  ). 
1  R.  V.  L.  HARTLEY,  U.  S.  Patent,  1218650. 


258  THERMIONIC  VACUUM  TUBE 

If  the  fraction  as  is  less  than  unity  the  series  is  convergent,  its  sum 
being  —  — ,  so  that  the  total  output  power  becomes: 

A  =  "( 


and  the  total  amplification  produced  is 


A^  1-s 
a      I  — as 


n  =  —  = 


Since  as<l  and  a  is  usually  large  compared  with  unity  for  any 
good  amplifier,  it  follows  that  s  can  be  neglected  in  comparison 
with  unity.  The  amplification  n  can  therefore  be  made  large  by 
making  as  nearly  equal  to  unity. 

If  as  should  be  equal  to  or  greater  than  unity,  the  above  series 
becomes  divergent  and  the  output  power  increases  without  limit 
until  checked  by  some  cause  determined  by  the  nature  of  the  cir- 
cuit characteristic.  The  output  power  becomes  independent  of 
the  original  input;  in  other  words,  the  system  produces  self- 
sustaining  oscillations.  If  an  inductive  circuit  is  used,  it  may 
happen  that  as  becomes  greater  than  unity  for  one  or  more  fre- 
quencies, depending  upon  the  impedance  and  phase  relations  in 
the  circuit,  with  the  result  that  the  system  oscillates  at  these 
frequencies. 

Hence,  in  order  to  use  this  scheme  for  amplifying  currents 
covering  a  wide  range  of  frequencies,  such  as  telephonic  currents, 
without  producing  sustained  oscillations,  it  is  advisable  to  use 
a  non-inductive  circuit.  Such  a  circuit  is  shown  in  Fig.  145. 
The  input  voltage  is  impressed  across  the  resistance  R\  and  the 
grid  of  tube  A  is  maintained  negative  in  the  usual  way  by  the 
battery  Eg.  In  the  circuit  as  shown  here  the  plate  batteries  Ej, 
are  connected  between  the  plates  and  the  output  resistances 
#2  and  Rz.  In  this  case,  therefore,  in  contrast  to  Fig.  144,  the 
succeeding  grids  tend  to  become  too  much  negative  due  to  the 
direct  space  currents  in  R%  and  Rz.  They  can  become  so 
much  negative  with  respect  to  their  filaments  that  they  may  choke 
down  the  space  currents.  It  is  therefore  necessary  to  insert  the 
grid  batteries  E'g  so  poled  as  to  reduce  the  negative  potentials 
of  the  grids  to  the  values  necessary  for  satisfactory  operation. 
The  plate  currents  pass  through  the  resistances  R2,  Rz  to  the 


THE  THERMIONIC  AMPLIFIER 


259 


common  point  of  the  circuit,  which  may  be  grounded.  The 
alternating  plate  current  of  the  third  tube  C,  which  may  be  sep- 
arated from  the  d-c.  by  a  condenser,  passes  through  the  recording 
device  S  which  the  system  is  supposed  to  operate  and  through  part 
of  the  resistance  R%  to  the  common  point. 

Consider  the  half  period  during  which  current  in  R\  flows  in 
the  direction  of  the  arrow.  This  makes  the  grid  of  the  tube  A  less 
negative  with  respect  to  its  filament.  The  output  electron  current 
in  R2  therefore  increases  in  the  direction  of  the  arrow.  This 
makes  the  grid  of  B  more  negative  with  respect  to  its  filament, 
so  that  the  space  current  in  B  decreases  or  increases  in  the  direction 
of  the  arrow  in  R%.  The  space  currents  in  successive  tubes  are 
therefore  180°  out  of  phase.  The  current  in  the  third  tube  flowing 


FIG.  145. 

through  S  is  in  phase  with  the  current  in  #2,  and  since  it  must 
flow  through  part  of  /fe  the  potential  of  the  grid  of  B  is  increased 
over  that  occasioned  by  the  space  current  in  tube  A.  It  will  be 
apparent  that  in  order  to  have  the  phase  relations  right,  the  output 
of  one  tube  must  be  returned  to  the  input  of  the  preceding  tube 
or  one  of  the  alternate  preceding  tubes.  The  amount  of  energy 
fed  back  can  be  controlled  by  varying  the  part  of  the  resistance  R% 
through  which  the  feed-back  current  flows. 

It  may  be  remarked  that  the  circuit  needs  careful  adjustment 
to  secure  markedly  increased  amplification  without  making  the 
system  "  sing/'  or  produce  sustained  oscillations.  If  an  inductive 
circuit  is  used  the  energy  fed  back  can  be  controlled  easily  by 
varying  the  coupling  between  the  output  and  input  coils.  In 


260  THERMIONIC  VACUUM  TUBE 

this  case  the  phase  relations  can  be  so  controlled  that  part  of 
the  output  of  a  tube  can  be  returned  in  proper  phase  to  its  own 
input.  We  have  then  simply  the  case  of  the  ordinary  "  feed-back 
circuit "  which  is  extensively  used  in  radio  reception. 

Multi-stage  amplifiers  frequently  have  a  tendency  to  sing. 
One  way  of  preventing  this  is  to  feed  part  of  the  output  energy 
back  to  the  input  in  opposite  phase. 

The  phase  relations  in  a  multi-stage  amplifier  set  furnish  a 
means  for  obtaining  distortionless  amplification.  To  illustrate 
this,  let  us  consider  the  non-inductive  two-stage  amplifier  shown  in 
Fig.  144,  and  let  us  suppose  that  the  two  tubes  A  and  B  are  alike. 
Suppose  also  that  the  resistances  r  and  TO  are  equal.  If  a  sinu- 
soidal voltage  e  be  impressed  on  the  grid  of  tube  A  the  output 
current  flowing  through  r  will  not  be  sinusoidal  unless  r  is  large. 
It  was  explained  in  Section  58  that  if  the  external  resistance  is 
equal  to  or  greater  than  the  plate  resistance  of  the  tube,  the 
characteristic  of  the  circuit  becomes  nearly  linear  and  therefore 
nearly  distortionless.  The  little  distortion  produced  by  the 
characteristic  being  not  quite  linear,  or  even  the  marked  dis- 
tortion produced  when  r  is  small,  can  be  further  reduced  by  using 
an  even  stage  amplifier  circuit.  Thus,  assuming  that  the  char- 
acteristic is  curved,  if  the  voltage  e\  impressed  on  the  input  circuit 
is  sinusoidal,  then  the  voltage  62  impressed  on  the  second  tube 
by  the  varying  current  flowing  in  r  will  be  lopsided,  as  shown  by 
the  curve  66'  of  Fig.  83,  page  167.  That  is,  when  the  grid  of  A 
becomes  less  negative  with  respect  to  its  filament,  a  greater  increase 
in  electron  current  is  produced  in  r  in  the  direction  of  the  arrow 
than  the  decrease  occasioned  by  the  grid  of  A  becoming  more 
negative  with  respect  to  its  filament.  The  potential  impressed 
on  the  grid  of  tube  B  is  therefore  lopsided,  but  it  is  180°  out  of 
phase  with  the  potential  on  the  grid  of  A.  The  increase  in  poten- 
tial of  grid  B  is  therefore  smaller  than  the  decrease  and  hence  if 
the  characteristic  of  tube  B  is  likewise  curved,  the  current  in  its 
output  resistance  TQ  will  be  nearly  sinusoidal.  It  is  obvious  that 
distortion  can  be  reduced  by  this  means  only  if  the  circuit  contains 
an  even  number  of  tubes. 

When  the  circuit  is  inductive,  the  phase  relations  are,  of  course 
not  so  simple.  In  such  cases  it  is  best  to  make  the  inductances 
in  the  output  circuits  of  the  tubes  sufficiently  large  to  straighten 
out  the  characteristic  to  the  desired  extent.  If  it  is  necessary 


THE  THERMIONIC  AMPLIFIER 


261 


to  use  a  low  impedance  in  the  output  circuit,  distortion  can  be 
reduced  by  means  of  the  " push-pull"  circuit  of  E.  H.  Colpitts, 
shown  in  Fig.  146.  It  is  supposed  that  the  output  circuit  must 
be  connected  to  a  device  of  low  impedance  in  which  case  the  dis- 
tortion produced  when  a  simple  circuit  is  used  may  be  considerable. 
The  way  in  which  the  "  push-pull "  circuit  eliminates  the  distor- 
tion can  be  understood  by  referring  to  Fig.  83.  The  shape  of  the 
output  current  wave  in  the  plate  circuit  of  the  tube  A  (Fig.  146) 
is  given  by  the  curve  bbf  (Fig.  83).  In  passing  through  the 
transformer  it  is  resolved  into  the  fundamental  ee  and  the  har- 
monic //.  The  wave  shape  of  the  current  in  tube  B  is  like  that  of 


FIG.  146. 

At  except  that,  since  the  potentials  on  the  grid  of  the  two  tubes 
are  180°  out  of  phase,  the  fundamental  currents  ee  in  the  two- 
plate  circuits  will  differ  by  180°,  while  the  harmonics  //  will  be  in 
phase.  It  will  therefore  be  seen,  by  referring  to  the  current 
directions  in  Fig.  146,  that  the  fundamentals  will  be  additive  in 
effect,  while  the  harmonics  will  neutralize  each  other. 

This  circuit  can  also  be  used  to  cut  out  the  fundamental  and 
transmit  only  the  harmonics,  by  reversing  the  primary  coil  of 
the  output  transformer  of  one  of  the  tubes.  This  was  suggested 
by  J.  R.  Carson,  for  the  purpose  of  modulation. 

It  will  be  apparent  that  a  circuit  like  that  shown  in  Fig.  146 
requires  that  the  two  tubes  be  alike  in  their  characteristics. 

Fig.  147  shows  an  arrangement  whereby  the  filaments  and 
plates  of  a  two-stage  amplifier  can  all  be  operated  from  a  single 


262 


THERMIONIC  VACUUM  TUBE 


source  of  voltage.  This  circuit  arrangement  is  of  advantage  when 
only  one  source  of  voltage  is  available,  such  as,  for  example,  the 
standard  city  mains  of  110  or  220  volts  direct  current.  The  con- 
densers Ci  and  €2  are  inserted  to  by-pass  the  alternating  currents 
in  the  plate  circuits  in  case  the  sections  of  the  resistance  R  between 
the  positive  terminal  of  the  voltage  supply  and  the  point  connecting 
the  plate,  should  become  so  large  as  to  cause  an  undesirable  waste 
of  a-c.  power  in  the  plate  circuit  of  the  tube. 

All  the  amplification  circuits  discussed  thus  far  are  unilateral; 
that  is,  they  transmit  and  amplify  currents  in  one  direction  only. 
For  most  purposes  unilateral  circuits  are  all  that  are  needed,  but 
for  amplification  of  currents  on  telephone  lines  the  amplifier  must 


FIG.  147. 

be  capable  of  transmitting  and  amplifying  currents  in  both  direc- 
tions, so  that  two-way  conversations  can  be  carried  on  over  the  line. 
The  circuit  that  is  commonly  used  in  telephone  practice,  using 
thermionic  tubes  as  amplifiers,  is  shown  in  Fig.  148. 1  It  is 
known  as  the  22-type  circuit  (two-way,  ^o-repeater  type).  The 
principle  of  this  type  of  repeater  circuit  was  disclosed  by  W.  L. 
Richards,  in  1895.2  As  applied  to  thermionic  amplifiers,  it  has" 
been  in  use  on  the  lines  of  the  Bell  Telephone  System  since  1913. 
The  principle  consists  in  balancing  each  of  the  two  lines  against 
an  artificial  line  or  balancing  network  having  an  impedance  equal 

1  B.  GHERARDI  and  F.  B.  JEWETT,  Proc.  A.I.E.E.,  Nov.,  1919,  p.  1297. 
The  reader  is  referred  to  this  paper  for  information  on  the  use  of  repeaters  on 
long  distance  telephone  lines. 

2  W.  L.  RICHARDS,  U.  S.  Patent  542657,  1895. 


THE  THERMIONIC  AMPLIFIER 


263 


to  that  of  the  line.  The  purpose  of  the  balancing  network  can  be 
understood  from  the  following :  The  currents  coming  from  Line  W, 
for  example,  are  branched  off  at  PI,  thus  furnishing  the  input  to 
tube  E,  the  amplified  output  of  which  is  obtained  in  the  output 
transformer  T\.  Similarly  the  amplified  currents  from  Line  E 
pass  through  the  output  transformer  T%.  Now,  if  the  system 
into  which  transformer  T\,  for  example,  feeds  were  not  symmetrical 
with  respect  to  the  points  PI,  the  amplified  current  in  T\  would 
cause  an  increased  input  to  be  impressed  on  tube  W;  this  in  turn 
increases  the  input  on  tube  E,  and  the  system  would  produce  sus- 
tained oscillations,  or  "  sing."  This  is  prevented  by  making 


the  impedance  of  the  balancing  network  equal  to  that  of  the 
line  to  which  it  is  connected,  thus  making  the  potentials  of  the 
points  PI  and  PI  independent  of  the  currents  in  the  corresponding 
output  coils  T2  and  T\.  This,  of  course,  results  in  a  division  of 
the  amplified  output  power,  one-half  becoming  available  and  the 
other  half  being  wasted  in  the  balancing  network. 

In  a  system  like  this  the  degree  of  amplification  obtainable 
without  impairing  the  quality  of  transmission  depends  largely 
on  the  accuracy  with  which  the  lines  can  be  balanced. 

Two-way  transmission  can  also  be  secured  with  a  single 
repeater,  by  means  of  a  circuit  arrangement  such  as  that  shown  in 


264 


THERMIONIC  VACUUM  TUBE 


Fig.  149.  This  circuit  is  known  as  the  "  21-type  repeater  circuit " 
(two-way,  one-repeater  circuit).  Instead  of  balancing  each  line 
with  an  artificial  network,  as  in  the  22-circuit,  two-way  trans- 
mission, with  the  21-circuit,  is  effected  by  balancing  the  two  lines 
directly  against  each  other.  Thus,  if  the  impedance  to  the 
right  of  P  is  equal  to  that  to  the  left  of  P,  then  the  potentials 
of  the  connecting  points  P  are  independent  of  the  amplified  output 
current.  But  if  the  impedances  of  these  lines  are  not  equal,  the 
amplified  current  in  the  output  will  impress  increased  potentials 


FIG.  149. 


on  the  grid  of  the  amplifier;  these  will  cause  further  amplification, 
the  amplification  becoming  cumulative,  thus  creating  sustained 
oscillations  or  "  singing." 

The  advantages  of  the  22-circuit  over  the  21-circuit  are  appa- 
rent: The  latter  requires  that  the  impedances  of  the  lines  (as 
measured  at  P)  leading  to  the  two  telephone  substations  between 
which  the  conversation  is  carried  on,  be  identical — a  condition 
which  cannot  always  readily  be  realized  in  practice.  In  the 
22-circuit,  the  two  lines  may  have  quite  different  impedances, 
the  requirements  being  then  that  the  balancing  networks  have 
different  impedances,  each  balancing  its  own  line.  The  repeater 
can  then  be  inserted  at  some  convenient  place  on  the  line,  which 


THE  THERMIONIC  AMPLIFIER  265 

need  not  be  midway  between  the  two  stations.  Furthermore,  in 
order  to  create  the  condition  of  singing  in  the  22-circuit,  it  is 
necessary,  as  can  readily  be  seen  from  Fig.  148,  that  both  lines  be 
unbalanced  simultaneously.  If  one  line  and  its  network  be  per- 
fectly balanced,  an  unbalance  in  the  other  will  not  cause  singing. 
The  22-circuit  is  therefore  inherently  more  stable  than  the  21- 
circuit. 

The  filters  are  inserted  to  pass  only  currents  lying  within  the 
telephone  frequency  range,  thus  preventing  the  passage  through 
the  repeaters  of  telegraph  and  other  signal  currents  that  may  be 
transmitted  over  the  same  metallic  circuits.  The  potentiometers 
are  inserted  to  adjust  the  amplification  to  the  desired  value. 


CHAPTER   VIII 
THE  VACUUM  TUBE  AS  AN  OSCILLATION   GENERATOR 

79.  Introductory.  Since  the  three-electrode  tube  can  operate 
as  an  amplifier,  the  energy  in  the  output  circuit  is  greater  than  that 
in  the  input  circuit.  Hence,  if  part  of  the  energy  in  the  output 
be  returned  to  the  input,  there  will  a  further  amplification  of 
energy  resulting  in  an  increased  output.  If  the  amplified  energy 
gets  back  to  the  input  in  sufficient  amount,  and  if  the  phase  rela- 
tions of  the  output  and  input  currents  are  right,  there  will  be  a 
constant  reamplification  and  feeding  back  of  energy  from  the 
output  to  the  input,  and  the  device  will  then  produce  sustained 
oscillations  without  it  being  necessary  to  supply  potential  varia- 
tions to  the  grid  by  external  means.  In  other  words,  the  device 
will  then  operate  as  an  oscillation  generator.  The  frequency  of 
the  oscillations  will  be  determined  by  the  constants  of  the  circuit, 
while  their  intensity  will  depend  on  the  amount  of  energy  fed 
back  to  the  input,  the  shape  of  the  characteristic  curve,  and 
the  rate  at  which  power  is  dissipated.  There  is  a  variety  of 
circuit  arrangements  whereby  this  can  be  done.  These  circuits 
can  all  be  divided  into  three  main  groups  in  which  part  of  the 
energy  in  the  output  is  returned  to  the  input:  (1)  by  resistance 
coupling,  such  as  is  explained,  for  example,  in  connection  with  Fig. 
145,  page  259;  (2)  inductive  coupling;  (3)  capacitative  coupling 
between  output  and  input. 

In  the  present  chapter  we  shall  briefly  discuss  the  conditions 
that  must  be  satisfied  in  order  to  use  the  three-element  tube  to 
produce  oscillations  of  a  definite  frequency  and  amplitude.  A 
large  amount  of  work  has  been  done  on  this  phase  of  the  subject, 
and  no  attempt  will  be  made  here  to  enter  into  a  full  discussion  of 
these  investigations.  It  's  believed  rather  that  an  explanation  of 
the  fundamental  principles  that  govern  the  production  of  sus- 
tained oscillations  with  the  three-electrode  tube,  and  an  indication 

266 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR         267 

of  the  more  important  results  that  can  be  obtained,  will  be  suf- 
ficient to  enable  anybody  who  understands  them  to  design  the 
circuits  that  may  be  necessary  for  any  particular  purpose. 

The  conditions  that  are  necessary  to  make  the  tube  act  as  an 
oscillation  generator  can  be  stated  briefly  as  follows: 

(1)  The  tube  must  be  capable  of  amplifying.     That  is,   it 
must  have  a  unilateral  impedance  which  is  occasioned  by  poten- 
tial variations  on  the  grid  producing  a  greater  effect  on  the  cur- 
rent in  the  plate  circuit  (output  circuit)  than  the  effect  produced 
on  the  current  in  the  grid  circuit  by  potential  variations  on  the 
plate.     This  property  is  generally  expressed,  as  was  explained  in 
the  previous  chapter,  by  stating  that  an  alternating  potential  eg, 
impressed  on  the  grid,  produces  an  E.M.F.  in  the  plate  circuit 
which  is  equal  to  nea  where  JJL  in  practice  is  generally  greater  than 
unity. 

(2)  Since  the  energy  in  the  output  is  greater  than  that  in  the 
input,  part  of  this  energy  can  be  returned  to  the  input,  but  in 
order  to  insure  a  reamplification  of  this  energy  it  is  necessary  to 
take  care  that  the  output  and  input  currents  are  in  phase. 

(3)  An  oscillation  circuit  must  be  attached  to  the  tube,  having 
inductance,  capacity  and  resistance  of  such  value  as  to  make 
the  tube  oscillate  with  the  desired  frequency.     These  quantities 
should,  for  best  operation,  also  be  so  adjusted  that  the  efficiency 
of  the  tube  as  an  oscillator  and  the  amount  of  power  delivered 
to  the  oscillation  circuit  are  as  large  as  possible. 

(4)  The  characteristic  of  the  tube  must  be  such  that  the  tube 
constants,  together  with  the  constants  in  the  oscillation  circuit, 
determine  the  amplitude  of  the  oscillations.     In  general  the  ampli- 
tude is  limited  by  the  factors  that  limit  the  flow  of  the  current 
through  the  tube.     These  factors  have  been  explained  in  Chap- 
ter IV. 

80.  Method  of  Procedure  for  the  Solution  of  the  Oscillation 
Equations.  The  complete  solution  of  the  oscillation  equations  of 
the  vacuum  tube  is  difficult  because  of  the  peculiar  shape  of  the 
tube  characteristics.  It  was  explained  in  Chapter  IV  that  the 
current-voltage  characteristic  is  such  that  when  taken  over  the 
whole  range  it  cannot  be  expressed  by  a  simple  equation.  For 
the  lower  part  of  the  characteristic,  where  the  effective  applied 
voltage  is  less  than  the  voltage  drop  in  the  filament,  the  current 
varies  approximately  as  the  f -power  of  the  voltage.  For 


268  THERMIONIC  VACUUM  TUBE 

high  effective  voltages,  the  exponent  of  the  voltage  decreases  as 
the  voltage  increases,  and  finally  approaches  zero  as  the  satura- 
tion current  is  approached.  When  using  the  tube  as  an  amplifier 
in  the  most  efficient  way,  we  operate  over  such  a  part  of  the 
characteristic  that  a  simple  quadratic  equation  can  be  used.  As 
was  explained  in  Chapter  VII,  special  precautions  are  taken  to 
make  the  characteristic  of  the  tube  and  circuit  as  nearly  linear 
as  possible.  Now,  when  using  the  tube  as  an  oscillation  generator 
we  seldom  make  use  of  a  restricted  portion  of  the  characteristic, 
but,  on  the  contrary,  the  plate  current  generally  oscillates  between 
zero  and  a  saturation  current  value,  and  it  is  therefore  difficult 
to  express  the  current  as  a  simple  function  of  the  applied  voltage. 
But  the  conditions  for  oscillation  can  be  derived  without  neces- 
sarily making  use  of  a  definite  characteristic  equation.  What 
we  shall  do  is  to  use  the  resistance  and  mutual  conductance  of 
the  tube  as  the  variable  parameters  in  terms  of  which  the  condi- 
tions for  oscillation  can  be  expressed,  and  then  see  how  these 
parameters  depend  on  the  characteristics  of  the  tube. 

An  expression  for  the  plate  resistance  of  the  tube  was  derived 
in  the  preceding  chapter,  for  the  general  case  in  which  the  current 
varies  as  the  nth  power  of  the  applied  voltage.  If  the  oscilla- 
tions are  extremely  small  then  the  resistance  is  given  by  the 
reciprocal  of  the  slope  of  the  plate  current-plate  potential  char- 
acteristic at  the  point  of  operation.  When  the  oscillations  are 
finite,  the  resistance  is  approximately  given  by  the  secant  joining 
the  points  of  maximum  and  minimum  current.  If  the  charac- 
teristic is  a  parabola,  this  secant  is  parallel  to  the  tangent  at  the 
point  of  zero  alternating  potential,  so  that  the  resistance  is  inde- 
pendent of  the  applied  alternating  voltage.  When  the  character- 
istic is  not  parabolic,  the  resistance  cannot  be  expressed  simply 
by  the  slope  of  the  tangent  at  the  point  of  zero  alternating  voltage, 
but  depends  on  the  magnitude  of  the  voltage.  Thus,  referring  to 
Fig.  150  it  will  be  seen  that  as  long  as  the  potential  variations  are 
less  than  AB,  the  resistance  is  practically  the  same  as  that  which 
obtains  for  infinitely  small  oscillations  around  the  point  A.  If, 
now,  the  maximum  value  of  the  potential  becomes  equal  to  AC, 
the  line  joining  C"  and  0'  is  not  parallel  to  the  tangent  at  A', 
but  has  a  smaller  slope.  We  can  take  the  slope  of  the  line  O'C' 
as  a  measure  of  the  resistance  when  the  current  variations  extend 
over  the  whole  region  O'A'C'. 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR         269 

The  same  considerations  apply  to  the  mutual  conductance 
which  can  be  taken  approximately  to  be  equal  to  the  slope  of  the 
line  joining  the  extreme  point  of  the  characteristic  over  which  the 
operation  takes  place.1 

The  problem  of  setting  up  the  conditions  for  oscillation, 
therefore,  reduces  to  the  solution  of  a  network  involving  the 
oscillation  circuit,  LCr,  a  fictitious  generator  giving  a  voltage 
equal  to  ^eg  and  a  resistance  rp,  as  defined  above.  By  adopting 
this  procedure,  we  do  not  entirely  ignore  the  curvature  of  the  char- 
acteristic. If  the  resistance  rp  were  not  dependent  on  the  intensity 


Anode 
FlG.  150. 

of  the  oscillations,  the  solution  of  the  network  involving  the  quan- 
tities enumerated  above  would  not  give  an  indication  of  the  ampli- 
tude of  the  oscillations;  however,  both  rp  and  the  mutual  con- 
ductance gm  are  dependent  on  the  intensity  of  the  oscillations. 
The  condition  for  oscillation  will  require  that  rp  and  gm  have  values 
lying  within  certain  limits,  and  since  their  values  depend  on  the 
extent  of  the  characteristic  over  which  operation  takes  place, 
the  amplitude  of  the  oscillations  will  be  determined  by  the  shape 
of  the  characteristic  and  the  constants  of  the  external  circuit. 

81.  Conditions   for    Oscillation   in   a    Two-element   Device. 
Let  us  first  consider  the  simple  case  of  a  device  containing  two 
1 L.  A.  HAZELTINE,  Proc.  I.R.E.,  Vol.  6,  p.  63,  1918. 


270 


THERMIONIC  VACUUM  TUBE 


electrodes  and  having  a  resistance  r,  and  see  what  are  the  con- 
ditions that  must  be  satisfied  in  order  that  this  device,  when 
connected  to  an  oscillation  circuit  may  produce  sustained  oscilla- 
tions. The  circuit  is  shown  diagrammatically  in  Fig.  151.  The 
device  is  supplied  with  a  direct  current  by  means  of  the  battery 
E,  through  the  choke  coil  Ch.  The  oscillation  circuit  is  repre- 
sented by  LC. 

The  condition  for  oscillation  in  such  a  circuit  can  be  obtained 

by  the  simple  process  of  setting  the  differential  equation  for  the 

circuit  and  equating  the  damping  factor  to  zero.     In  doing  so 

we  need,  of  course,  only  to  consider  the  a-c.  circuits,  that  is,  the 

Ch 


(JUOd  f  —  . 

fn 

^ 

•v; 

-=- 

i 

^K      ff 

rp 

t 

7                VP            J 

MV 
•HM 

41 

rl_, 

c 

FIG.  151. 

two  branches  I  and  II.     Thus,  summing  the  electromotive  forces 
for  circuit  I,  we  get 


where  p  =    . 

For  branch  II  we  have 


(1) 


(2) 


Putting  the  value  of  i\  given  by  equation  (I)  into  (2)  we  get: 


This  is  an  equation  of  the  well-known  form  from  which  it  follows 
directly  that  in  order  that  the  current  i  shall  be  oscillatory,  the 
coefficient  of  the  linear  term  must  be  zero.  That  is: 


Cr 


(4) 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR        271 

From  this  we  see  that  in  order  to  obtain  sustained  oscillations 
from  a  device  having  only  two  electrodes,  it  is  necessary  that  the 
device  shall  have  a  negative  resistance.  Examples  of  negative 
resistances  have  already  been  given  in  the  previous  pages.  Thus 
an  arc  may  have  a  negative  resistance,  its  characteristic  being 
of  the  form  shown  by  the  curve  AB  in  Fig.  34.  The  resistance 
is  given  by. the  slope  of  the  characteristic  and  this  is  a  negative 
quantity  for  a  characteristic  of  the  kind  shown.  Fig.  16,  page  48, 
shows  another  characteristic  which  over  a  region  ABC  has  a  nega- 
tive slope  and  is  obtained  as  the  result  of  the  emission  of  electrons 
from  metals  under  the  impact  of  electrons.  Such  characteristics 
as  these  are  sometimes  referred  to  as  "  falling  characteristics." 

The  thermionic  valve  does  not  show  a  falling  characteristic 
like  the  curve  A  B  of  Fig.  34,  when  it  is  sufficiently  well  evacuated 
to  prevent  the  effects  of  ionization  by  collision  from  appreciably 
influencing  the  discharge.  The  characteristics  of  thermionic 
valves  are  those  given  and  discussed  in  the  previous  chapters  and 
it  will  be  seen  that  for  such  devices  the  resistance  k  always  posi- 
tive. It  is,  therefore,  impossible  to  obtain  sustained  oscillations 
from  a  well-evacuated  thermionic  valve  containing  only  two 
electrodes.  If  such  a  device  contains  an  appreciable  amount  of 
gas  during  the  operation,  the  characteristic  becomes  unsteady 
and  sometimes,  especially  at  the  higher  voltages,  exhibits  regions 
over  which  the  slope  is  negative,  and  when  operated  over  that 
region  it  is,  of  course,  possible  to  obtain  sustained  oscillations. 
The  condition  which  makes  this  possible  in  a  two-electrode  device 
is  unfortunately  due  to  the  cause  which  makes  such  a  device 
unsatisfactory;  namely,  the  presence  of  too  much  gas  in  the 
device,  thus  causing  unsteadiness  of  the  discharge  and  making 
reproducibility  practically  impossible.  If  a  controlling  electrode 
or  grid  be  added  to  the  two  electrodes  of  a  valve,  the  operation 
becomes  different  and  we  now  have  a  device  which  can  produce 
sustained  oscillations  with  facility  while  at  the  same  time  satisfying 
all  the  conditions  that  are  necessary  to  secure  satisfactory  opera- 
tion in  every  respect,  namely,  freedom  of  gas  with  consequent 
steadiness  and  reproducibility  of  results,  and  comparatively  long 
life. 

82.  Condition  for  Oscillation  for  Three-Electrode  Tube. 
Let  us  now  consider  a  circuit  like  that  shown  in  Fig.  152.  This 
is  one  of  a  large  number  of  possible  oscillation  circuits.  It  is 


272 


THERMIONIC  VACUUM  TUBE 


chosen  here  to  exemplify  the  manner  in  which  oscillations  can  be 
produced  with  an  audion  because  this  type  of  circuit  lends  itself 
most  readily  to  mathematical  solution.  The  plate  current  is 
supplied  by  the  battery  Eb  through  a  choke  coil.  The  oscillation 
circuit  proper  is  represented  by  CL^r.  On  account  of  the  mutual 
inductance  M  between  the  coils  L\  and  Z/2,  current  variations  in 
Z/2  cause  potential  variations  to  be  impressed  on  the  grid.  The 
oscillation  circuit  CrZ/2  is  practically  non-reactive  at  the  oscillation 
frequency.  A  current  in  the  plate  circuit  establishes  an  E.M.F. 
in  the  circuit  1/2(7, 90°  out  of  phase  with  the  plate  current  (assuming 
that  r  is  very  small.  Since  the  oscillation  circuit  is  non-reactive, 
the  oscillation  current  is  in  phase  with  the  E.M.F.  in  this  circuit 


FIG.  152. 

and  this  current  induces  an  E.M.F.  in  the  grid  coil  LI,  90°  out  of 
phase.  The  potential  variations  impressed  on  the  grid  by  the 
reaction  of  the  plate  circuit  on  the  grid  circuit  are  therefore  in 
phase  with  the  plate  current  variations. 

In  order  to  obtain  an  expression  for  the  condition  that  must 
be  satisfied  by  such  a  circuit  to  produce  sustained  oscillations  we 
shall  make  the  following  assumptions  which  can  be  realized  in 
practice,  although  this  may  not  generally  be  the  case.  The  approx- 
imation is,  however,  sufficiently  good  to  give  an  indication  of 
the  quantities  involved  and  the  values  that  they  should  have  in 
order  to  make  it  possible  for  such  a  circuit  to  act  as  an  oscillation 
generator.  We  shall  assume  (1)  that  the  grid  is  at  all  times  main- 
tained sufficiently  negative  with  respect  to  the  filament  to  prevent 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR        273 

any  convection  current  from  flowing  between  filament  and  grid. 
(2)  The  capacities  between  the  electrodes  of  the  tube  will  be 
taken  to  be  sufficiently  small  to  be  neglected.  The  tube  is  there- 
fore assumed  to  be  a  perfectly  unilateral  device;  that  is,  there  is  no 
reaction  of  the  plate  circuit  on  the  grid  circuit  except  through 
inductance  of  the  coils  LI  and  £2.  (3)  The  oscillation  circuit 
will  be  regarded  as  non-reactive  at  the  frequency  of  oscillation, 
or  at  least  the  angle  can  be  taken  to  be  so  small  that  its  effect 
on  the  phase  relations  of  the  circuit  can  for  the  present  be  neglected. 
This  is  usually  very  nearly  the  case  in  most  circuits,  and  circuits 
can  be  designed  for  which  this  is  always  true. 

A  current  i  in  the  coil  LI  will  induce  an  E.M.F.  eg  in  coil  LI 
given  by 


-Mdi- 

i  —  *«  ~r.  — 


dt 


Mpi. 


Making  use  of  the  theorem  that  a  potential  e0  impressed  on  the 
grid  introduces  an  E.M.F.  equal  to  ^ec  in  the  plate  circuit,  we 
obtain  directly  for  the  driving  E.M.F.  in  the  plate  circuit: 


ljL6g  = 


(5) 


We  can,  therefore,  simplify  this  circuit  and  give  it  the  equivalent 
form  shown  in  Fig.  153,  where  rp  represents  the  plate  resistance 
of  the  tube,  the  points  P  and  F  representing  the  connections  to 
the  plate  and  the  filament.  The  generator  G  is  a  fictitious  gen- 
erator included  in  the  circuit  to  represent  the  effect  of  the  grid 


274  THERMIONIC  VACUUM  TUBE 

potential  variations  on  the  plate  circuit.  This  generator,  there- 
fore, gives  an  E.M.F.  equal  to  the  value  given  by  equation  (5). 
The  solution  of  the  circuit  now  becomes  extremely  simple,  involving 
only  the  solution  of  the  Kirchhoff  equations  for  the  simple  net- 
work to  the  right  of  PF.  The  circuit  on  the  left  of  PF  is  a  d-c. 
circuit  and  need  not  be  considered  because  we  are  concerned  now 
only  with  the  a-c.  values. 

Summing  the  E.M.F.'s  in  branch  I,  we  get 

uMpi  =  rp(i+ii)  +ri+L2pi, 
which  gives 

(6) 


For  branch  II,  we  have 

L2<[Pi+rpi  =  *±  ........     (7) 

Substituting  the  value  for  i\  into  equation  (7)  we  obtain: 


This  is  a  simple  differential  equation  of  the  well-known  form: 
p2i+Api+Bi=Q, 

from  which  we  obtain  directly  that  the  condition  that  must  be 
complied  with  to  produce  sustained  oscillations  in  circuit  II, 
is  that  A  must  be  equal  to  zero,  and  putting  p  =  jco  where  j  is  the 
imaginary  unit  V  -1,  the  frequency  of  oscillation  is 


Substituting  the  values  of  B  and  A=0  from  equation  (8),  we 
find  that  the  frequency  of  oscillation  is  given  by 


(9) 

J^'4\j 

and  the  condition  for  oscillation  is 

u.M    L2 
r>=fr-Cr <1(» 


VACUUM  TUBE  AS  AN  OSCILLATION   GENERATOR       275 


We  have  seen  above  that  the  condition  for  oscillation  in  the 
case  of  a  two- electrode  device  is  that  the  resistance  of  the  device 

must  be  negative  and  equal  to  -^-.     This  is  the  effective  resistance 

(_y  7* 

of  the  oscillation  circuit,  so  that  when  it  is  added  to  the  equal  and 
opposite  resistance  of  the  device  the  total  resistance  of  the  circuit, 
and  therefore  the  damping,  is  zero.  From  equation  (10)  we  see 
that  in  the  case  of  a  three-electrode  tube  the  resistance  of  the  tube 

need  not  be  negative  as  long  as  the  first  term  -^—  is  large  enough. 

\jT 

This  term  involves  the  amplification  constant  /*  and  therefore  indi- 
cates directly  that  the  ability  of  the  audion  to  produce  oscilla- 
tions lies  in  its  amplifying  property. 

In  order  to  give  an  interpretation  to  this  condition  (equation 
(10)  let  us  write  it  in  the  form: 


Cr  1 


or 


Cr 


jr4& 


(11) 


It  will  be  recognized  that  gm  =  —  is  the  mutual  conductance  of 

Tp 

the  tube  as  defined  in  the  preceding  chapter.  For  very  small 
oscillations  the  mutual  conductance  is  given  by  the  slope  of  the 
plate  current  grid  potential  characteristic,  while  for  large  oscil- 
lations it  can  be  taken  to  be  approximately  equal  to  the  slope 
of  the  line  joining  the  points  of  maximum  and  minimum  current 
on  the  characteristic.  Now,  it  will  be  recognized  that  as  the 
intensity  of  the  oscillations  increases,  the  slope  of  this  line  becomes 
less  and  less.  Equation  (11),  on  the  other  hand,  states  that  for 
oscillations  to  be  sustained  the  mutual  conductance  must  be 
greater  than  a  quantity  involving  the  constants  of  the  external 
circuit.  The  right-hand  side  of  equation  (11)  is  also  of  the 
dimensions  of  a  conductance  and  can  also  be  represented  by  a  line 
having  a  slope  depending  on  the  values  of  these  constants.  Sup- 
pose this  line  has  a  definite  slope  given  by  OA,  Fig.  154.  The 
oscillations  will,  therefore,  increase  in  intensity,  the  current  vary- 
ing over  a  greater  and  greater  range  of  the  characteristic1  until  the 


276 


THERMIONIC  VACUUM  TUBE 


mutual  conductance  as  given  by  the  line  BC  joining  the  points  of 
maximum  and  minimum  current  becomes  parallel  to  OA. 

If  the  mutual  inductance  between  the  plate  and  grid  coils 
were  decreased,  the  slope  of  the  line  representing  the  right-hand 
side  of  equation  (11)  would  increase,  say,  to  OA',  and  then  the 
oscillations  would  be  weaker,  the  plate  current  varying  over  such 
a  range  that  the  mutual  conductance  is  equal  to  the  slope  of  the 
line  OA'. 


Grid.  Volfs     -  0  + 

FIG.  154. 


Whether  or  not  the  tube  will  oscillate  depends  not  only  on 
the  coupling  between  the  output  and  the  input  coils,  but  also 
on  a  number  of  other  quantities.  One  of  the  important  quan- 
tities is  the  amplification  constant  /*•  Fig.  155  shows  how  /z  in- 
fluences the  operation  of  the  device  as  an  oscillator.  The  line  OB 


gives  —  as  a  function  of  /*,  and  CD  gives  the  expression 


of  equation  (11)  as  a  function  of  /-t.     We  shall  refer  to  this  quantity 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR       277 

as  go.  This  equation  states  that  gm  must  be  at  least  equal  to  go; 
hence,  for  the  constants  of  the  circuit  chosen  in  this  particular 
case,  all  values  of  M  lying  to  the  left  of  the  broken  line  are  impos- 
sible values. 

The  effect  of  the  plate  voltage  can  be  shown  in  a  similar  way. 
It  follows,  for  example,  from  the  considerations  given  in  Chapter 


10 


FIG.  155. 

VII,  that  the  mutual  conductance  is  approximately  proportional 
to  the  plate  potential,  provided  the  filament  temperature  is  high 
enough  to  insure  that  by  increasing  the  plate  potential  we  do  not 

enter  into  the  saturation  region.     We  can,  therefore,  replace  — 

rv 

and  rp  by  EP  and  some  arbitrary  constants.     We  then  obtain 


278 


THERMIONIC  VACUUM  TUBE 


an  expression  gm  =  go  where  gm  is  directly  proportional  to  Ep  and  go 
is  a  linear  function  of  Ep.  These  relations  when  plotted  as  shown 
in  Fig.  156  intersect  at  the  point  A.  The  condition  for  oscillation 
is  that  gm  must  be  at  least  as  large  as  go.  We  see,  therefore,  that 
the  tube  will  not  oscillate  until  the  plate  voltage  reaches  a  certain 
minimum  value  which  is  fixed  if  the  other  quantities,  such  as  the 
coupling,  etc.,  are  fixed. 

These  considerations  show  that  it  is  desirable  to  make  the 


50 


20  30  40 

Plate  Volts 

FIG.  156. 


50 


mutual  conductance  of  the  tube  as  large  as  possible.  This  was 
also  found  to  be  the  case  when  using  the  tube  as  an  amplifier. 
(See  equation  (36),  Chapter  VII). 

The  oscillation  frequency  as  given  by  equation  (9)  is  deter- 
mined not  entirely  by  the  inductance  and  capacity  in  the  oscilla- 
tion circuit,  but  depends  also  on  the  plate  resistance  of  the  tube 
and  the  resistance  in  the  oscillation  circuit.  Since,  however,  the 

ratio  --is  usually  very  small,  the  frequency  can  generally  be 
r 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR        279 

taken  to  be  very  closely  equal  to  that  given  by  the  simple  oscilla- 
tion circuit;  namely, 


(12) 


It  will  be  recognized  that  the  solution  of  the  circuit  shown  in 
Fig.  153  does  not  indicate  directly  what  the  amplitude  of  the  oscil- 
lations is.  This  quantity  is,  however,  determined  indirectly  by 
the  condition  for  oscillation.  On  account  of  the  curvature  of  the 
characteristic,  the  mutual  conductance  of  the  tube  decreases  as 
the  amplitude  of  the  oscillations  increases,  in  the  manner  explained 
above,  until  the  mutual  conductance  reaches  its  minimum  value. 
The  amplitude  of  the  oscillations  in  the  plate  circuit  can  then  be 
determined  from  this  limiting  value  and  the  characteristic  of  the 
tube.  Usually,  however,  such  a  determination  is  not  necessary. 

83.  Relation  between  Mutual  Conductance  of  Tube  and  that 

of  Plate  Circuit.    In  the  above  equations,  —  represents  the  mutual 

r-p 

conductance  of  the  tube  itself.  This  is  also  the  mutual  conduct- 
ance of  the  plate  circuit,  provided  the  external  impedance  in  the 
plate  circuit  is  negligibly  small  compared  with  the  resistance  of  the 
tube.  When  this  is  not  the  case,  the  dynamic  characteristic  of 
the  plate  circuit  does  not  coincide  with  the  characteristic  of  the 
tube  itself,  but  differs  from  it  to  an  extent  depending  on  the  rela- 
tive magnitudes  of  the  external  impedance  and  the  plate  resistance. 
If  the  external  circuit  is  non-reactive,  the  dynamic  characteristic 
of  the  plate  circuit  is  the  curve  of  noC  shown  in  Fig.  86.  If  the 
external  circuit  is  reactive,  the  dynamic  characteristic  of  the  plate 
circuit  takes  the  form  of  the  loop  such  as  that  shown  in  Fig.  88. 
In  this  case  the  quantity  concerned  is  not  a  pure  conductance  but 
a  complex  quantity,  and  what  we  have  to  deal  with  then  is  the 
mutual  admittance.  In  most  oscillation  circuits,  however,  the 
reactance  is  so  small  at  the  oscillation  frequency  in  comparison 
with  the  total  resistance  that  the  angle  can  generally  be  neglected. 
It  can  readily  be  seen  that  the  mutual  conductance  of  the  cir- 
cuit is  less  than  that  of  the  tube  alone,  because  when  the  current 
in  the  external  circuit  is  increased  by  an  increase  in  the  potential 
of  the  grid,  the  voltage  drop  in  the  external  impedance  causes  a 
decrease  in  the  plate  potential,  so  that  the  resultant  increase  in 
plate  current  is  less  than  would  be  the  case  if  the  external  imped- 


280  THERMIONIC  VACUUM  TUBE 

ance  were  zero.  The  relation  between  these  two  mutual  con- 
ductances can  be  obtained  as  follows:  The  alternating  plate  cur- 
rent is  given  by 


Putting  Z0=r0+jx0  we  get 


P,Qo 
/*+  M    V 


where  Y'm  =  g'm—jb  is  the  mutual  admittance  of  the  plate  circuit. 
Generally  the  imaginary  component  is  small  in  comparison  with 
the  resistance  component,  so  that  we  can  use  the  simple  equation: 

4-=1+-°.      ........     (14) 

9  m      Qm       M 

This  relationship  can  be  expressed  in  a  somewhat  different  form. 
Since  ipro=ep,  we  get  directly  from  equation  (13): 


The  condition  for  oscillation  can  also  be  expressed  in  terms  of 
the  mutual  conductance  of  the  plate  circuit  instead  of  the  mutual 
conductance  of  the  tube  itself.  This  was,  for  example,  done  by 
Hazeltine.1  The  quantity  g  in  Hazeltine's  equations  is  not  the 
mutual  conductance  of  the  tube,  but  the  mutual  conductance  of 
the  plate  circuit. 

84.  Phase  Relations.  The  phase  relations  that  exist  in 
vacuum  tube  oscillator  circuits  have  been  investigated  by  Heising 
and  explained  with  the  help  of  vector  diagrams.2  We  shall  not 
discuss  this  phase  of  the  subject  beyond  what  is  necessary  for  an 
understanding  of  the  fundamental  phenomena  of  such  circuits. 
The  main  condition  is  that  the  plate  current  and  grid  potential 
must  be  as  nearly  in  phase  as  possible.  The  phase  relations  be- 
tween the  various  quantities  are  shown  in  Fig.  157  and  can  be 
explained  with  reference  to  Fig.  152. 

Ib  represents  the  steady  direct  current  supplied  by  the  battery 
Eb  through  the  choke  coil.  We  can  regard  this  current  as  constant, 

1  L.  A.  HAZELTINE,  R.I.E.,  Vol.  6,  p.  63,  1918. 

2  R.  A.  HEISING,  Journal  of  A.I.E.E.,  Vol.  39,  p.  365,  1920. 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR        281 


although  in  actual  practice  it  is  only  approximately  so  unless  the 
choke  coil  has  a  very  large  inductance.  IP  represents  the  instan- 
taneous value  of  the  plate  current  and  /  the  instantaneous  value  of 
the  current  in  the  branch  containing  C  and  Z/2  in  parallel.  This 
current  multiplied  by  the  instantaneous  plate-filament  voltage 
and  integrated  over  a  complete  cycle,  represents  the  a-c.  power 
supplied  by  the  tube.  As  much  power  is  drawn  from  the  tube  as 


Grid  Potential 


Anode  Current 


Anode  Potential ' 


Output  Current 


FIG.  157. 

is  dissipated  in  the  oscillation  circuit  when  the  steady  condition 
is  reached. 

Referring  to  Fig.  157,  the  lines  marked  0  represent  the  ordinates 
of  zero  voltages  and  currents.  The  grid  is  maintained  at  a  nega- 
tive potential  Ec.  When  the  alternating  grid  potential  is  zero, 
the  plate  current  is  equal  to  7&.  When  the  grid  potential  oscillates, 
as  indicated,  the  plate  current  oscillates  in  phase  with  the 
grid  potential.  The  plate  potential  oscillates  around  the  mean 
value  Ei,,  but  is  180°  out  of  phase  with  the  grid  potential  if  the 


282 


THERMIONIC  VACUUM  TUBE 


external  circuit  is  non-reactive.  The  current  7  in  the  branch  cir- 
cuit is  the  difference  between  direct  current  Ib  drawn  from  the 
battery  and  the  plate  current  IP.  It  is  therefore  180°  out  of 
phase  with  the  plate  current  and  oscillates  around  zero.  We  have 
assumed  that  the  grid  always  remains  negative  with  respect  to 
the  negative  end  of  the  filament.  If  the  grid  becomes  positive 
during  a  part  of  the  cycle,  it  takes  current  which  generally  means  a 
loss  of  power  occasioned  by  heat  dissipation  in  the  grid  circuit. 

On  account  of  the  curvature  of  the  characteristic,  the  current 
wave  in  the  plate  circuit,  due  to  a  sinusoidal  voltage  impressed  on 
the  grid  circuit,  is  not  a  pure  sinusoid  but  is  distorted.  This 
introduces  harmonics.  They  can,  however,  be  effectively  tuned 


ii 


„-*— ^ 

* 


FIG.  158. 

out  in  the  oscillation  circuit  so  that  most  of  the  energy  in  the 
oscillation  circuit  will  be  due  to  the  fundamental.  It  must  be 
recognized  that  the  harmonics  cause  a  waste  of  power.  These 
considerations  apply  in  general  to  the  fundamental,  the  effect  of 
harmonics  being  neglected. 

85.  Colpitts  and  Hartley  Circuits.  The  circuit  shown  in  Fig. 
152  is  only  one  of  a  large  number  that  can  be  used  with  a  vacuum 
tube  oscillator.  It  was  chosen  there  for  its  simplicity,  although 
it  is  not  the  most  commonly  used  type  of  circuit.  Two  circuits 
that  are  frequently  used  are  those  shown  in  Figs.  158  and  159, 
known  as  the  Colpitts  and  Hartley  circuits,  respectively.  The 
main  difference  between  these  circuits  is  that  in  the  one  the  coup- 
ling between  output  and  input  circuits  is  capacitive  and  in  the 
other  it  is  mainly  inductive.  If  we  neglect  the  effect  of  the  elec- 
trostatic capacities  between  the  electrodes  of  the  tube,  the  oscil- 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR        283 

lation  circuits  are  C\C2L,  for  the  Colpitts  circuit,  and  LiL2C,  for 
the  Hartley  circuit.  The  effect  of  the  inter-electrode  capacities 
will  be  considered  below. 

The  conditions  for  oscillation  for  these  circuits  have  been 
given  by  Hazeltine,  Heising  l  and  others.  Taking,  for  example, 
the  case  of  the  Hartley  circuit,  the  condition  for  oscillation  can  be 
expressed  by: 

gm=  (L2+M)Lu(L!+M)  -  (L2+M)]>     *     '     '     (16) 

where  gm  =  mutual  conductance  of  the  tube, 

M  =  mutual  inductance  between  L\  and  L2, 


FIG.  159. 

From  this  equation  it  follows  that  there  is  a  certain  relation 
between  the  voltages  established  in  the  plate  and  grid  coils,  which 
makes  the  tube  oscillate  most  readily.  Since  the  conditions  for 
oscillation  state  that  the  right-hand  side  of  equation  (16)  must 
not  be  greater  than  #„,.  it  follows  that  the  tube  will  oscillate  most 
readily  when  this  expression  is  a  minimum. 

Putting 


—  =n,  we  find 
eg 

1  Loc.  cit, 


284 


THERMIONIC  VACUUM  TUBE 


where  A;  is  a  constant. 


This  is  a  minimum  for 


MCr  (l+n)2 
Qm~  L   njL-n 


M+2' 


(17) 

(18) 


For  tubes  having  a  high  value  of  M,  therefore,  LI  should  be  approx- 
imately L2.  If  M  is  low,  on  the  other  hand,  the  best  condition  can 
necessitate  making  Z/2  considerably  smaller  than  LI. 

86.  Tuned  Grid-circuit  Oscillator.  This  type  of  circuit 
which  is  commonly  used  in  the  reception  of  radio  signals,  is  shown 
in  Fig.  160.  If  it  is  assumed,  as  before,  that  the  grid  is  maintained 


FIG.  160. 


at  a  sufficiently  high  negative  potential  to  insure  that  there  is  no 
appreciable  convection  between  filament  and  grid,  the  condition 
for  oscillation  for  this  circuit  can  also  be  easily  obtained.  The 
potential  eg  applied  to  the  grid  is  given  by 


(19) 


and  the  electromotive  force  induced  in  the  plate  circuit  through 
the  tube,  on  account  of  the  effect  of  the  grid  potential  on  the 
current  is  nea.  There  is  another  electromotive  force  induced 
in  the  plate  circuit,  namely,  Mpi,  and  is  due  to  the  mutual  react- 
ance of  the  grid  circuit  on  the  plate  circuit  through  the  coils  LI 
and  L2.  The  electromotive  force  induced  on  the  oscillation  cir- 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR        285 

cuit,  due  to  the  current  ip  in  the  plate  circuit,  is  Mpip.     Equating 
these  E.M.F/s  in  the  circuits,  we  get  for  circuit  /: 


,      .....     (20) 
and  for  circuit  II  : 


(21) 


Eliminating  ip  from  these  two  equations,  the  equation  for  i  becomes  : 
(LiL2  -  M2)p*i  +  (r*Li  +rL2)p2i+  (rrPC+L2  - 


0,     .      (22) 
which  is  of  the  form 

p*i+Ap*i+Bpi+Di  =  0. 

This  is  a  cubic  equation  and  has  one  real  and  two  complex  roots. 
The  condition  which  makes  the  damping  zero  is  D  =  AB.     That  is: 


In  most  circuits  rL2  can  be  neglected  in  comparison  with  rpL\ 
With  this  approximation  the  condition  for  oscillation  becomes: 


« 


The  right-hand  side  of  this  equation  contains  two  terms,  one  of 
which  is  directly  proportional  to  M,  and  the  other  inversely  pro- 
portional to  M  .  There  appears,  therefore,  to  be  an  optimum  value 
for  the  mutual  inductance  between  the  input  and  output  which 
makes  gm  a  minimum.1 

87.  Effect  of  Inter-electrode  Capacities  —  Parasitic  Circuits. 
We  have  assumed  in  the  above  that  there  is  no  reaction  of  the  plate 
circuit  on  the  grid  circuit  through  the  tube  itself.  In  some  types 
of  circuits  the  capacities  between  the  electrodes  cause  the  circuits 
to  behave  differently,  from  what  is  to  be  expected.  The  simple 
circuit  shown  in  Fig.  160  can,  for  example,  be  drawn  in  the  manner 
shown  in  Fig.  161,  where  the  capacities  between  the  electrodes  of 
the  tube  are  indicated  Ci,  C2  and  €3.  Such  a  circuit,  therefore, 
has  more  than  one  degree  of  freedom,  a  number  of  oscillation  cir' 
1  S.  BALLANTINE,  Proc.  I.R.E.,  Vol.  7,  p.  159. 


286 


THERMIONIC  VACUUM  TUBE 


cults  being  added  to  the  main  oscillation  circuit  CL\.    Of  these 
parasitic  circuits,  the  most  important  one  in  the  diagram  shown 


is  the  circuit  formed  by  the  capacity  Cs  between  grid  and  plate, 
and  the  inductance  Z/i  and  Lz  in  series,  the  total  inductance  being 


Frequency 

FIG.  162. 


Li+L,2+2M.  The  effect  of  the  capacity  Ca  is  to  make  the  fre- 
quency of  osciUation  different  from  that  which  would  be  obtained 
from  a  simple  circuit  CL\.  The  reactance-frequency  curve  of  the 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR        287 

circuit  CLi  is  shown  by  the  curve  marked  X\  (Fig.  162).  For 
frequencies  lower  than  that  corresponding  to  the  point  A,  this  cir- 
cuit has  a  positive  or  inductive  reactance.  The  effective  react- 
ance, due  to  the  coil  LI  and  its  coupling  with  the  oscillation  circuit 
Z/iC,  is  given  by  the  curve  X%.  At  frequencies  below  A,  the  total 
reactance  is  inductive,  and  oscillations  will  occur  at  such  a  fre- 
quency that  the  inductive  reactance  is  equal  to  the  capacitive 
reactance  due  to  the  capacity  €3  between  grid  and  plate.  The 
oscillation  frequency  is,  therefore,  that  corresponding  to  the  point 
F  instead  of  the  point  A,  as  would  be  the  case  if  the  effect  of  the 
grid-plate  capacity  were  negligible.  This  circuit,  therefore,  be- 
haves somewhat  like  a  Hartley  circuit  in  that  the  plate  coil  Z/2, 
and  the  oscillation  circuit  L\C,  together,  act  like  an  inductance  in 


r9 


FIG.  163. 

parallel  with  a  capacity.  In  the  Hartley  circuit,  the  capacity 
between  grid  and  plate  is  simply  in  parallel  with  the  oscillation 
circuit  capacity  C.  This  circuit  is,  therefore,  more  suitable  for 
use  at  high  frequencies. 

88.  Regeneration.  The  effect  of  the  inter-electrode  capacities 
can  cause  a  tube  to  produce  oscillations  even  when  there  is  no 
mutual  inductance  M  between  the  output  and  input  coils.  It 
was  explained  in  Chapter  VII,  Sections  69—71,  that  on  account 
of  these  capacities  there  is  an  effective  impedance  between  fila- 
ment and  grid,  which  depends  not  only  on  the  capacities  between 
the  electrodes  but  also  on  the  constants  of  the  output  circuit. 
This  impedance  can  generally  be  represented  by  a  resistance  rg 
and  a  reactance  due  to  the  effective  input  capacity  Cg.  The  input 
circuit  can,  therefore,  be  drawn  as  shown  in  Fig.  163.  The  im- 


288  THERMIONIC  VACUUM  TUBE 

pedance  of  the  circuit  formed  by  C  in  parallel  with  rc  and  Ca  in 
series  is: 


The  real  component  r  is 

(26) 


The  first  term  of  the  denominator  in  this  equation  is  usually 
negligibly  small  compared  with  the  second,  so  that  the  total 
resistance  of  the  circuit  is: 


If  oscillations  are  impressed  on  this  circuit,  the  rate  at  which  they 
would  die  out  depends,  of  course,  on  the  value  of  the  total  resist- 
ance. If  rg  is  negative,  the  total  resistance  will  be  reduced;  that 
is,  there  will  be  a  smaller  consumption  of  power  in  the  input  cir- 
cuit and  the  tube  will  give  a  greater  amount  of  amplification. 
If  ra  is  negative  and  the  effective  resistance  to  the  right  of  AB  is 
equal  to  the  resistance  ri,  i.e.,  if  the  total  resistance  of  the  circuit 
is  zero,  the  circuit  will  produce  sustained  oscillations.  We  can 
take  expression  (27)  as  a  measure  of  the  damping,  6,  due  to  the 
resistance  in  the  circuit.  The  increase  in  amplification,  due  to  a 
reduction  in  this  resistance  results  in  the  effect  that  is  sometimes 
referred  to  as  "  regeneration."  A  measure  of  the  regenerating 

effect  is  given  by  -.     Now,  it  was  shown  in  Chapter  VII  that  ra 
o 

is  positive  when  the  external  plate  circuit  is  non-reactive  or  con- 
tains only  capacitive  reactance.  If,  on  the  other  hand,  the 
reactance  in  the  plate  circuit  is  inductive  and  the  angle  of  the 
impedance  in  the  plate  circuit  is  large  enough,  then  ra  is  negative. 
In  Fig.  164  are  plotted  curves  showing  the  relation  between  the 
regenerative  effect  and  the  ratio  of  the  external  impedance  in  the 
plate  circuit  to  the  plate  resistance.  The  values  of  ra  and  C0, 
used  in  computing  these  curves,  were  obtained  from  equations 

(54)  and  (56)  of  Chapter  VII.     The  quantity  T  was  computed 

5 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR        289 

with  the  values  so  found  and  with  arbitrarily  assumed  values  of 
rij  as  indicated  in  the  curves.  The  curve  for  ri  =  5.4  ohms 
stretches  to  infinity,  indicating  that  over  the  range  of  the  ratio 

rp 

—  from  about  .8  to  about  1.2,  the  tube  produces  sustained  oscil- 

Tv 

lations,  due  to  the  reaction  of  the  plate  circuit  on  the  grid  circuit 
through  the  electrostatic  capacities  of  the  tube. 


.3.0 


An  interesting  result  shown  by  these  curves  is  that  the  max- 
imum regenerating  effect  is  obtained  when  the  external  impedance 
in  the  plate  circuit  is  equal  to  the  plate  resistance.  This,  it 
will  be  remembered,  is  also  the  condition  for  maximum  power 
amplification  derived  in  Chapter  VII,  for  the  case  of  the  simple 
amplifier. 


290 


THERMIONIC  VACUUM  TUBE 


According  to  expression  (27)  the  regenerating  effect  becomes 
greater  the  smaller  the  capacity  C  in  the  oscillating  circuit. 

J.  M.  Miller  1  has  computed  curves  giving  the  signal  strength 
as  a  function  of  the  inductance  in  the  plate  circuit.  These  curves 
are  similar  to  those  shown  in  Fig.  164,  except  that  they  are  more 
symmetrical.  They  have  the  same  general  form  as  a  curve  giving 
experimental  results  published  by  Armstrong.2 

89.  Complex  and  Coupled  Circuits— Meissner  Circuit.  Com- 
plex circuits  can  be  reduced  to  simple  circuits  by  the  addition  of 
the  reactances  of  the  separate  branches.  Thus,  the  circuit  shown 
in  Fig.  160  constitutes  a  complex  circuit  if  the  capacity  between 
grid  and  plate  becomes  effective  in  determining  the  frequency  of 
the  oscillations.  The  reactances  of  the  branches  I  and  II  are 
indicated  by  the  curves  X\  and  X%  of  Fig.  162,  while  the  reactance 


FIG.  165. 

due  to  the  capacity  €3  between  grid  and  plate  is  indicated  by  X%. 
The  frequency  of  the  oscillations  is  determined  by  the  value  of 
these  reactances  which  makes  the  total  reactance  of  the  circuit 
zero. 

In  general,  a  complex  circuit  such  as  that  shown  in  Fig.  165 
can  be  regarded  as  a  simple  circuit  in  which  the  oscillation  circuits 
LiCi  and  LiCi  act  as  inductances  or  capacities,  according  as  the 
reactance  between  grid  and  plate  is  capacitive  or  inductive.  The 
reactance-frequency  curve  of  a  parallel  oscillation  circuit  like  L\C\ 
has  a  shape  such  as  the  curve  X\  in  Fig.  162.  Now,  if  the  react- 
ance between  grid  and  plate  is  capacitive  or  negative,  the  total 

1  J.  M.  MILLER,  Bureau  of  Standards  Bulletin  351. 

2  E.  H.  ARMSTRONG,  Proc.  I.R.E.,  Vol.  3,  p.  220,  1915. 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR        291 


reactance  can  only  become  zero  at  a  frequency  F,  which  is  lower' 
than  the  natural  oscillation  frequency  A  of  the  simple  parallel 
circuit  LiCi  or  L^C^  From  the  curve  it  is  seen  that  at  frequencies 
below  A  the  reactance  of  the  simple  circuit  L\C\  is  inductive 
(positive),  so  that  the  complex  circuit  shown  in  Fig.  165  can  be 
represented  by  the  simple  circuit  of  Fig.  166.  This  is  a  Hartley 
circuit.  If,  on  the  other  hand,  the  reactance  between  plate  and 
grid  were  inductive,  the  frequency  of  oscillation  of  the  circuit 
will  be  such  as  to  make  the  sum  of  this  inductive  reactance  and 
the  reactances  of  the  branch  circuits  I  and  II  equal  to  zero.  This 
would  require  that  the  reactances  of  these  branch  circuits  be 
capacitive  and  therefore  the  frequency  of  oscillation  will  be  higher 
than  the  natural  frequency  of  the  circuits  I  and  II  separately. 


o 

0> 

o 
o 
o 


FIG.  166. 

In  this  case  the  total  circuit  reduces  to  one  like  that  shown  in 
Fig.  166,  except  that  instead  of  the  two  inductances,  we  have  two 
capacities,  and  instead  of  the  capacity  €3,  we  have  an  inductance. 
In  other  words,  the  complex  circuit  is  reduced  to  a  simple  Colpitts 
circuit. 

Coupled  circuits  can  be  treated  in  much  the  same  manner. 
The  Meissner  circuit  shown  in  Fig.  167  is  an  example  of  a  coupled 
circuit.1  The  two  oscillation  circuits  are  LiZ^Ca  and  LC.  The 
effects  that  can  be  obtained  with  such  a  circuit  by  varying  the 
constants  of  the  oscillation  circuits  have  been  discussed  by 
Heising  in  the  paper  cited  above.  The  reactance-frequency  curves 
of  such  a  circuit  are  shown  in  Fig.  168,  where  X\  represents  the 
reactance  of  the  circuit  LiZ^Cs,  and  X%  the  reactance  due  to 
1  A.  MEISSNER,  "  Electrician,"  Vol.  73,  p.  702,  1914. 


292 


THERMIONIC  VACUUM  TUBE 


coupling  with  the  oscillation  circuit  LC.  The  sum  of  the  two 
reactances  is  indicated  by  the  curve  X.  It  is  seen  that  there  are 
three  frequencies  for  which  the  total  reactance  is  zero.  The  tube 
will  not  oscillate  at  the  frequency  F  because  this  represents  an 
unstable  condition,  but  it  can  oscillate  at  the  two  frequencies 
Fi  and  7^2.  Usually,  however,  it  oscillates  at  only  one  frequency. 
By  suitably  adjusting  the  coupling  between  the  oscillation  circuit 
LC,  or  choosing  the  constants  of  the  circuits,  the  reactance-fre- 
quency curve  of  the  combination  can  take  such  a  form  that  the 
three  frequencies  practically  merge  into  one.  This  can  be  done 
by  making  the  coupling  loose,  or  by  making  the  total  inductance 


i 


^mmm} 

••         f 


\M 


FIG.  167. 

large  compared  to  €3.    This  is  usually  the  case  with  most 
tubes  when  the  desired  frequency  is  not  very  high. 

90.  Circuits  Comprising  a-c.  and  d-c.  Branches.  The  circuits 
shown  above  indicate  only  the  a-c.  branches.  These  are  the  only 
branches  that  need  to  be  considered  in  determining  the  conditions 
for  oscillation  and  the  frequency.  In  practice  we  also  need  d-c. 
sources  of  power  supply,  and  it  is  often  necessary  to  separate  the 
a-c.  and  d-c.  circuits.  This  can  be  done  readily  by  applying  the 
simple  and  well-known  rule  to 'separate  the  d-c.  from  the  a-c. 
branches  by  means  of  inductances  and  capacities.  In  doing  so, 
however,  it  is  necessary  to  adjust  these  inductances  and  capacities 
to  such  values  that  they  do  not  appreciably  influence  the  behavior 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR         293 

of  the  oscillation  circuit  proper,   or  introduce  parasitic  circuits 
that  would  result  in  a  loss  of  power. 

Fig.  169  shows,  as  an  example,  the  Hartley  circuit  as  it  is  com- 
monly used  in  practice.  The  plate  battery  Eb  is  inserted  directly 
in  the  circuit  connecting  the  plate  to  the  inductance  L^  It  is 
usually  not  necessary  to  separate  the  direct  and  alternating  cur- 
rent in  this  branch  of  the  circuit.  The  capacity  Cs  and  resistance 
Rs  are  used  here  instead  of  a  battery  to  maintain  the  grid  at  an 


FIG.  168. 


appropriate  negative  potential  with  respect  to  the  filament.  This 
means  of  maintaining  the  grid  negative  operates  only  when  there 
is  a  convection  current  between  filament  and  grid;  that  is,  when 
the  grid  becomes  positive  during  part  of  the  time  that  the  a-c. 
potential  on  the  grid  is  positive.  When  the  grid  becomes  positive 
it  attracts  electrons,  and  current  flows  through  the  resistance  Rs. 
During  the  rest  of  the  cycle  there  is  no  flow  of  electrons  from  fila- 
ment to  grid.  There  is,  therefore,  established  a  rectified  current 


294 


THERMIONIC  VACUUM  TUBE 


through  the  resistance  Rs,  and  this  lowers  the  potential  of  the  grid 
with  respect  to  the  filament. 

4 


FIG.  169. 


Fig.  170  shows  a  Colpitts  circuit  as  it  can  be  used  in  practice. 
In  this  case  the  resistance  Rs  is  replaced  by  a  choke  coil  Chi. 
The  alternating  and  direct  current  in  the  plate  circuit  are  sep- 
arated by  means  of  the  choke  coil  Chz  and  the  capacity  C&.  The 


FIG.  170. 

inductance  of  this  choke  coil  is  usually  chosen  as  high  as  possible, 
or  at  least  so  high  that  its  impedance  is  several  times  the  plate 
resistance  of  the  tube.  The  capacities  Cs  and  C6  are  chosen 
sufficiently  large  so  that  they  do  not  appreciably  affect  the  opera- 
tion of  the  oscillation  circuit  LC\Cz. 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR        295 

91.  Effect  of  Grid  Current.  In  deriving  the  conditions  for 
oscillation  above,  it  was  assumed  that  the  grid  is  at  all  tunes 
maintained  sufficiently  negative  with  respect  to  the  filament  to 
prevent  any  convection  current' from  flowing  between  filament  and 
grid.  In  practice  this  is  usually  not  the  case.  The  grid  usually 
becomes  positive  to  an  extent  depending  on  the  adjustments  of 
the  circuit  constants.  Thus,  when  using  the  condenser  and 
resistance  to  maintain  the  grid  negative,  as  shown  in  Fig.  169, 
the  grid  must  become  positive  during  part  of  the  cycle.  The 
rectified  current  through  Rs  maintains  the  grid  at  a  steady  nega- 
tive potential,  and  the  value  of  this  potential  will  be  greater  the 
greater  the  grid  current  becomes.  Hence,  the  fraction  of  a  period 
during  which  grid  current  flows,  that  is,  the  amount  of  grid  cur- 
rent, will  be  determined  by  the  rate  at  which  it  leaks  off  through 
the  resistance  Rs.  Now,  the  grid  and  plate  potentials  are  approx- 
imately 180°  out  of  phase.  This  is  shown,  for  example,  in  Fig. 
157,  which  was  drawn  for  the  case  in  which  the  plate  is  connected 
to  a  non-reactive  circuit.  The  amount  of  current  flowing  to  the 
grid  depends  not  only  on  the  grid  potential  but  also  on  the  potential 
of  the  plate.  The  higher  the  plate  potential  the  more  readily  will 
the  electrons  be  drawn  through  the  openings  of  the  grid,  and 
the  smaller  will  be  the  grid  current.  But  if  the  plate  becomes 
less  positive  at  the  same  time  that  the  grid  becomes  more  positive 
(as  shown  in  Fig.  157),  there  is  a  tendency  for  the  grid  current  to 
become  much  greater,  and  if  the  potential  variations  impressed 
on  the  grid  become  so  great  that  the  maximum  positive  grid 
potential  becomes  equal  to  or  perhaps  greater  than  the  simul- 
taneous minimum  plate  potential,  then  the  grid  can  rob  the  plate 
of  so  much  current  that  the  characteristic  representing  the  plate 
current  as  a  function  of  the  grid  potential  becomes  apparently 
saturated  at  a  current  value  which  is  lower  than  the  actual  satura- 
tion current  at  the  temperature  of  the  filament.  This  effect  is 
shown  in  Fig.  171  which  represents  the  static  characteristics  of  the 
tube  for  various  fixed  values  of  the  plate  potential.  When  the 
tube  operates  in  an  oscillation  circuit,  which  is  so  adjusted  that 
the  reactance  in  the  plate  circuit  is  practically  zero,  the  dynamic 
characteristic  of  the  plate  circuit  is  represented  by  the  curve  AOB. 
The  normal  grid  potential,  when  the  tube  does  not  oscillate,  is 
represented  by  the  value  Ec.  When  oscillating,  the  plate  poten- 
tial decreases  when  the  grid  potential  increases  and  vice  versa, 


296 


THERMIONIC  VACUUM  TUBE 


so  that  the  characteristic  is  given  by  A  OB  instead  of  the  static 
characteristic  A'O'B'.  At  B  the  plate  and  grid  potentials  become 
comparable,  and  the  plate  current  begins  to  decrease.  This 
plate  current  is  less  than  the  emission  current;  that  is,  the  current 
represented  by  the  total  number  of  electrons  leaving  the  filament. 
The  point  B  represents  the  maximum  potential  that  the  grid  can 
acquire  without  causing  too  much  waste  of  power. 

The  mutual  conductance  of  the  plate  circuit  can  be  repre- 
sented by  the  slope  of  the  straight  line  joining  A  and  B.  The 
instantaneous  value  of  this  quantity  is  zero  at  B  and  A,  and  has  a 


B' 


A'          Grid  Potential 
FIG.  171. 

maximum  value  at  0,  but  the  mean  value  which  is  determined 
by  the  integrated  slope  of  the  curve  is  finite. 

92.  Output  Power.  The  value  of  the  grid  potential  at 
which  the  bend  B  in  Fig.  171  occurs,  depends  on  the  resistance 
of  the  external  plate  circuit.  The  current  in  this  resistance  r0 
causes  a  potential  drop  which  reduces  the  potential  on  the  plate, 
since  the  voltage  of  the  plate  battery  remains  constant.  The 
larger  this  resistance  the  greater  will  be  the  decrease  in  the  plate 
potential  when  the  current  in  the  plate  circuit  increases,  and  the 
sooner  will  the  bend  in  the  characteristic  occur.  Also,  the  greater 
the  external  resistance,  the  smaller  will  be  the  slope  of  the  dynamic 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR       297 


characteristic.  Fig.  172  shows  the  dynamic  characteristics  for  a 
number  of  different  values  of  the  external  plate  resistance  ro. 
The  characteristic  OB  is  obtained  for  the  largest,  and  OE  for  the 
smallest  external  resistance.  If  the  tube  operates,  for  example, 
in  a  circuit  like  that  shown  in  Fig.  152,  the  impedance  of  the  parallel 
circuit  L2r  and  C  is  given  by 

,  .coL2(l-co2CL2)-coCr2 


Z  = 


(l-co2CL2)2+co2C2r2  ~~     (l-co2CL2)-co2C2r 


(28) 


The  resistance  or  real  component  of  this  impedance  at  the 

L2 

resonance   frequency   is   given   by   ro=—.     The    characteristics 

Or 


Gnd 

FIG.  172. 

OB,  OC,  etc.,  correspond  to  different  values  of  this  resistance. 
The  maximum  power  output  in  each  case  is  obtained  when  the 
grid  potential  rises  up  to  the  value  indicated  at  the  bend.  If, 
now,  the  output  for  each  of  these  resistances  ro  be  plotted  as  a 
function  of  this  resistance,  we  obtain  a  curve  such  as  that  shown 
in  Fig.  173,  which  shows  a  maximum  for  a  particular  value  of  the 
external  resistance  ro.  This  resistance  is  equal  to  the  plate  resist- 
ance of  the  tube.  From  the  maximum  value  given  in  Fig.  173,  the 
ratio  of  the  inductance  to  the  capacity  of  the  oscillation  circuit 
can  be  determined,  which  gives  the  maximum  output  power. 
In  general,  maximum  output  power  does  not  necessarily  mean 


THERMIONIC  VACUUM  TUBE 


J 


maximum  efficiency.     This  will  become  evident  from  considera- 
tions in  the  next  section. 

93.  Efficiency. — The  efficiency  of  the  oscillator  can  be  ex- 
pressed by  the  ratio  of  the  power  supplied  to  the  oscillation  circui  t 
to  the  power  drawn  from  the  source  of  plate  voltage  (battery, 
generator,  etc.).  Strictly  speaking,  the  overall  efficiency  should 
take  account  also  of  the  power  expended  in  heating  the  filament. 
In  high  power  oscillators  this  power  can  usually  be  neglected, 
as  will  become  evident  from  the  following  considerations.  Of 


External  Resistance 
FIG.  173. 

the  power  drawn  from  the  source  of  plate  voltage  E»,  part  is  dissi- 
pated at  the  plate  and  serves  no  useful  purpose.  This  we  shall 
refer  to  as  Pp.  The  remainder,  P0,  of  the  power  drawn  from  the 
plate  battery,  is  delivered  to  the  oscillation  circuit.  If  we  neglect 
the  power  expended  in  heating  the  filament,  the  efficiency  can  be 
expressed  as: 

Efficiency  =  ^£=- 


where  Pb  represents  the  power  supplied  by  the  plate  battery  and 
is  equal  to  the  product  of  the  direct  current  Ib  and  direct  voltage 
Eb  in  the  plate  circuit. 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR       299 

J 

For  a  fixed  value  of  the  efficiency,  the  power  drawn  from  the 

plate  battery  is  a  measure  of  the  power  supplied  to  the  oscillation 
circuit.  This  can  be  increased  either  by  increasing  the  plate  cur- 
rent or  the  plate  voltage.  In  the  former  case,  the  filament  area 
or  the  filament  temperature,  or  both,  must  be  increased  and  then 
the  power  expended  in  heating  the  filament  may  become  com- 
parable with,  or  even  greater  than,  the  power  in  the  plate  circuit. 
If,  on  the  other  hand,  the  power  in  the  plate  circuit  be  increased 
by  increasing  the  plate  battery  voltage,  instead  of  the  current, 
the  power  expended  in  heating  the  filament  becomes  relatively 
smaller  and  smaller.  It  will  be  evident  that  even  if  the  saturation 
current  obtained  from  the  filament  remains  relatively  small,  the 
power  can  be  increased  to  almost  any  desired  value  by  sufficiently 
raising  the  voltage  of  the  plate  current  supply.  Of  course,  in 
doing  so  the  operating  point  of  the  characteristic  rises  to  higher 
current  values  and  may  even  fall  on  the  saturation  part  of  the 
characteristic,  but  this  can  be  prevented  by  increasing  the  nega- 
tive potential  on  the  grid  or  increasing  the  value  of  the  amplifica- 
tion constant  ju.  As  was  explained  in  Chapter  VII,  the  plate 
potential  necessary  to  give  any  chosen  value  of  plate  current 
increases  as  ju  is  increased.  The  limitation  to  increase  in  power  by 
this  means  is  not  inherent  in  the  tube,  but  is  determined  almost 
entirely  by  the  available  source  of  high  voltage  direct  current. 
The  plate  voltage  is  usually  supplied  by  one  of  three  means: 
Battery,  d-c.  generator,  or  vacuum  tube  rectifying  system,  such 
as  those  explained  in  Chapter  VI.  Batteries  are  costly  and  are 
seldom  used  for  high-power  tubes,  while  high-voltage  d-c.  gene- 
rators are  at  present  inefficient.  Vacuum  tube  rectifying  systems 
have  been  used  successfully  and  are  capable  of  giving  very  high 
voltages  but  care  must  be  taken  to  smooth  out  the  rectified  cur- 
rent wave,  and  the  extent  to  which  it  is  smoothed  out  by  means  of 
filters,  for  example,  depends  on  the  resistance  of  the  load  in  its 
output.  If  satisfactory  d-c.  generators  could  be  made  to  give 
from  10,000  to  20,000  volts,  it  would  be  possible  to  get  several 
kilowatts  output  power  from  tubes  that  are  relatively  inexpensive 
and  simple  to  make. 

In  considering  the  efficiency,  we  shall  neglect  the  power  dis- 
sipated in  heating  the  filament. 

Let  Ej,,  Eg,  Ip  be  the  instantaneous  plate  and  grid  potentials 
and  plate  current,  and  Eb,  EC)  h,  the  corresponding  d-c.  values 


300  THERMIONIC  VACUUM  TUBE 

when  the  alternating  components  are  zero.  Let  ep,  etc.,  be  the 
corresponding  R.M.S.  values,  and  e'P  the  maximum  a-c.  values. 
From  the  curves  shown  in  Fig.  157,  it  follows  that 

Ep  =  Eb+e'psmpt,     .......     (29) 

and  referring  to  Fig.  152,  we  see  that 

Ip  =  Ib-I  =  Ib-i' sinpt (30) 

The  power  dissipated  at  the  plate  is  given  by 

1   C^ 
Pp  =  2~  I     Eplpdt (31) 

or,  putting  in  the  values  from  the  above  two  equations  and  integrat- 
ing: 

6  V1    —P^_pn  /Q0\ 

—^r—*i>    *o \3*) 


The  power  dissipated  at  the  plate  is  therefore  equal  to  the 
power  supplied  by  the  plate  battery  minus  the  power  supplied 
to  the  oscillation  circuit,  and  the  efficiency  is 


(33) 


This  is  never  greater  than  50  per  cent  and  becomes  equal  to  50 
per  cent  if  the  plate  current  oscillates  over  the  whole  range  of  the 
characteristic  and  the  maximum  value  of  the  alternating  plate 
potential  is  such  as  to  reduce  the  plate  potential  to  zero  at  the 
moment  when  the  grid  has  its  maximum  positive  potential.  Under 
these  conditions  e'p=Eb,  and  i' '  =  /&. 

This  expression  was  derived  on  the  assumption  that  the  values 
of  Ep  and  Ip  are  always  within  the  limits  of  the  characteristic. 
There  is,  however,  another  way  in  which  the  tube  can  be  operated, 
which  gives  higher  efficiency.  This  can  be  done  by  so  propor- 
tioning the  plate  and  grid  potentials  that  the  plate  current  flows 
only  during  a  small  part  of  the  cycle.  Taking,  for  example,  the 
case  in  which  the  plate  and  grid  potentials  are  so  adjusted  that 
the  operating  point  does  not  lie  on  the  characteristic,  but  is  sit- 
uated beyond  the  intersection  of  the  characteristic  with  the  axis 
of  the  grid  potential  as  indicated  at  A  (Fig.  174) ,  it  will  be  seen 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR       301 

that  the  plate  current  flows  only  during  a  part  of  the  half  period 
during  which  the  a-c.  component  of  the  grid  potential  is  positive. 
During  the  time  that  the  plate  current  is  zero,  the  power  dissipated 
at  the  plate  is,  of  course,  zero.  When  current  flows  to  the  plate 
the  potential  of  the  plate  decreases  on  account  of  the  voltage 
established  in  the  external  resistance.  If  the  current  could 
become  so  large  that  the  potential  of  the  plate  is  reduced  to 


Grid  Pofenf-ial 
FIG.  174. 


practically  zero,  the  power  dissipated  at  the  plate  is  again  nearly 
zero,  so  that  the  total  power  dissipated  at  the  plate  becomes  very 
small.  In  the  extreme  case  in  which  the  current  at  a  given  grid 
potential  rises  suddenly  to  such  a  high  value  that  the  plate  poten- 
tial is  almost  immediately  reduced  to  zero,  the  total  power  dis- 
sipated at  the  plate  would  become  zero  and  then  the  efficiency 
would  be  100  per  cent.  This  is,  of  course,  a  theoretical  limit 
which  never  obtains  in  practice.  The  plate  potential  would  hardly 
ever  drop  down  to  zero  because  when  it  drops  so  low  as  to  become 
about  equal  to  the  simultaneously  occurring  instantaneous  value  of 


302 


THERMIONIC  VACUUM  TUBE 


the  grid  potential,  the  electrons  coming  from  the  filament  would 
be  diverted  to  the  grid  and  the  plate  current  would  .decrease,  so 
that  in  general  the  plate  potential  would  not  at  any  time  become 
lower  than  the  grid  potential. 

The  increase  in  efficiency,  when  operating  the  tube  somewhat 


I  Power  Dissipated 
at  Plate 


Output fowzr 


FIG,  175. 

in  the  manner  described  above,  can  be  explained  with  reference 
to  the  curves  in  Fig.  175.  These  curves  are  drawn  for  the  con- 
ditions that  current  to  the  plate  flows  only  during  part  of  a  cycle, 
and  that  the  plate  potential  when  the  current  is  a  maximum  is 
reduced  to  a  small  value.  For  such  an  irregular  set  of  curves 
the  simple  analysis  given  above,  and  which  led  to  equation  (33), 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR        303 

cannot  readily  be  applied.     But  the  efficiency  can  still  be  expressed 
by  the  equation 


^—- ^—  (34) 


JEjdt+jEpIjli 

In  Fig.  175,  the  horizontal  dotted  lines  represent  the  axes  of 
zero  values.  The  plate  current  is  represented  by  the  curve  just 
above  the  plate  potential  curve.  The  output  current  is  repre- 
sented by  /,  and  indicated  by  the  curve  below  the  plate  potential 
curve.  This  current  has  a  mean  value  indicated  by  the  dotted 
line.  The  plate  current  IP  flows  only  during  the  period  indiaated 
by  MN.  During  this  period  the  plate  potential  drops  and  the 
power  dissipated  at  the  plate,  which  is  given  by  J  Eplpdi,  is  rep- 
resented by  the  area  A.  The  output  power,  on  the  other  hand, 
is  given  by  J  Epldt,  and  is  represented  by  the  difference  between 
the  shaded  areas  B  and  C.  This  power,  for  the  conditions  chosen, 
is  greater  than  the  power  dissipated  at  the  plate,  the  efficiency  for 
the  values  chosen  here  being  about  70  per  cent. 

Fig.  176  shows  oscillograms  of  the  currents  and  voltages  taken 
under  such  conditions  that  the  plate  current  remains  zero  during 
about  half  of  a  complete  cycle.  For  this  I  am  indebted  to  Mr. 
J.  C.  Schelling.  The  photograph  shows  two  sets  of  curves  taken 
on  the  same  film,  /o  represents  the  current  in  the  oscillation  cir- 
cuit and  is  90°  out  of  phase  with  the  plate  current  IP.  The  grid 
current  Ig  is  in  phase  with  the  plate  current  and  the  grid  potential, 
and  these  quantities  are  nearly  180°  out  of  phase  with  the  plate 
potential  Ep. 

J.  H.  Morecroft l  has  computed  the  efficiency  for  a  number  of 
assumed  shapes  of  the  plate  current  wave.  These  curves  are 
shown  in  Fig.  177.  Below  the  figure  are  indicated  the  power 
dissipated  at  the  plate,  the  output  power  and  the  efficiency  for 
each  of  the  assumed  shapes  of  the  plate  current  waves.  In  the 
last  case,  where  the  current  wave  is  assumed  to  be  rectangular, 
the  efficiency  rises  to  a  value  of  82  per  cent.  This  represents  the 
best  condition  as  far  as  output  power  and  efficiency  are  con- 
cerned, but  in  general  the  output  power  decreases  as  the  efficiency 
increases  because  this  assumed  shape  of  wave  is  not  obtained  in 

1  Transactions  of  A.I.E.E.,  1919. 


304 


THERMIONIC  VACUUM  TUBE 


Ig.  Grid 


Current 


Jo,  Oscilla- 
tion Current 


Ip,  Plate 


Current 


Ep.  Plate 

Potential 


Eg,  Grid 


Potential 


Ip,  Plate 


Current 


FIG.  176. 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR       305 

practice.     In  practice  it  is  therefore  necessary  to  compromise 
between  efficiency  and  output  power. 

The  importance  of  high  efficiency  becomes  apparent  when 
considering  that  the  limiting  factor  in  a  tube  is  the  amount  of 
power  that  can  be  dissipated  at  the  plate.  This  is  limited  by  the 
factors  discussed  in  Section  31.  If  we  write  rj  for  the  efficiency, 
we  see  from  equation  (34)  that  the  output  power  is  given  by 


Since  PP  is  a  fixed  quantity  for  a  given  tube,  the  output  power 
could  be  made  very  large  by  making  77  large.     For  example,  if 


Efficiency  -39% 


Efficiency  =62%, 


FIG.  177, 


the  efficiency  could  be  as  high  as  90  per  cent,  and  if  the  plate  is 
capable  of  dissipating,  say,  500  watts,  then  the  output  power 
would  be  about  4.5  kw.  This  would  not  require  a  very  large  tube. 
The  total  area  of  the  plates  that  would  be  necessary  for  a  dissi- 
pation of  about  500  watts  ranges  from  about  50  to  100  sq.  cm., 
depending  on  the  material  used  for  the  plates.  When  using  a  tube 
in  this  manner,  it  is  necessary  to  remember  that  when  the  tube 
stops  oscillating,  for  example,  when  the  oscillation  circuit  is 
detuned,  the  total  power  supplied  by  the  plate  battery  will  be  dis- 


306 


THERMIONIC  VACUUM  TUBE 


sipated  at  the  plate  and  may  cause  the  liberation  of  too  much  gas, 
or  even  melt  the  plates.  It  is  therefore  necessary  to  insure  that 
whenever  the  oscillations  should  stop,  the  plate  battery  be  imme- 
diately cut  out,  or  its  voltage  be  sufficiently  reduced. 

94.  Method  of  Adjusting  Coupling  between  Output  and  Input. 
In  order  to  obtain  the  best  operation  with  a  tube,  it  is  necessary 
to  adjust  properly  the  coupling  between  the  output  and  input 
circuits.  In  most  circuits  this  is  readily  done  by  making  use  of 
any  of  well-known  means  of  changing  the  mutual  reactance. 
In  some  circuits,  however,  changing  the  coupling  also  changes 
the  oscillation  frequency.  Fig.  178  shows,  for  example,  a  Col- 


FIG.  178. 

pitts  circuit;  that  is,  a  circuit  in  which  the  coupling  between 
input  and  output  is  capacitive.  The  coupling  is  changed  by 
changing  the  condenser  €2,  but  it  will  be  seen  that  this  at  the 
same  time  changes  the  frequency  of  the  oscillation  circuit  LCiCz. 
This  is  usually  taken  care  of  by  inserting  another  condenser  Ca, 
the  capacity  of  which  is  then  so  adjusted  as  to  bring  the  fre- 
quency back  to  its  original  value.  Such  a  circuit  requires  two 
adjustments  when  it  is  necessary  to  change  the  coupling  while 
keeping  the  frequency  constant. 

A  circuit  which  avoids  this  double  adjustment  has  been 
described  by  R.  A.  Heising.1  This  circuit  is  shown  in  Fig.  179. 
The  oscillation  circuit  is  given  by  LC\C2  and  the  mutual  reac- 
tance between  the  output  and  input  is  varied  by  varying  the  con- 
tact Q.  This  adds  an  inductive  reactance  to  the  capacitive 

1  Loc.  cit. 


VACUUM  TUBE  AS  AN'  OSCILLATION  GENERATOR        307 


reactance,  thereby  changing  the  mutual  reactance  between  the 
output  and  input  circuits.  It  will  be  recognized  that  if  the  plate 
is  connected  to  the  point  Q',  the  circuit  is  the  same  as  that  shown 
in  Fig.  178,  with  the  capacity  Ca  omitted.  This  means  of  adjust- 
ing the  coupling  does  not  appreciably  change  the  oscillation  fre- 
quency. 

95.  Influence  of  the  Operating  Parameters  on  the  Behavior 
of  the  Oscillator.  It  will  be  realized  that  there  are  a  large  number 
of  factors  that  determine  the  operation  of  a  vacuum  tube  oscillator. 
The  most  important  of  these  factors  are  the  filament  current, 
d-c.  plate  and  grid  potentials,  plate  and  grid  coupling  and  oscillation 
circuit  resistance.  When  it  is  a  mere  matter  of  obtaining  an 

C  * 


FIG.  179. 

alternating  current  by  means  of  the  vacuum  tube,  very  few 
adjustments  will  serve  the  purpose.  •  It  will  usually  be  found  that 
the  tube  starts  oscillating  immediately  on  closing  the  plate  and 
filament  circuits.  If  it  fails  to  oscillate  a  slight  increase  in  fila- 
ment current  or  plate  voltage,  or  both,  will  set  the  tube  oscillating. 
If,  on  the  other  hand,  it  is  desired  to  obtain  maximum  power  out- 
put at  the  maximum  efficiency  consistent  with  it,  the  adjustments 
have  to  be  made  carefully,  but  with  a  little  practice  the  whole 
operation  reduces  to  a  simple  one.  Some  of  the  operating  para- 
meters are  fixed  by  the  limits  of  the  tube  and  circuit.  For  exam- 
ple, the  tube  may  be  designed  to  operate  on  a  certain  range  of 
filament  current  and  plate  battery  voltage.  This  automatically 
fixes  two  parameters.  The  manner  in  which  the  behavior  of  the 
oscillator  is  influenced  by  these  various  parameters  can  be  explained 
with  reference  to  the  following  diagrams.  These  represent  in  a 


308  THERMIONIC  VACUUM  TUBE 

general  way,  what  can  be  expected  with  commonly  used  types  of 
tubes.  The  nature  of  these  curves  could  be  expected  to  vary  some- 
what with  different  types  of  tubes.1  One  of  the  most  important 
variables  is  the  filament  current.  The  influence  of  the  filament 
current  on  the  operation  of  the  tube  can  be  understood  by  lef erring 
to  Figs.  17  and  18,  that  were  discussed  in  the  beginning  of  Chapter 
IV.  Fig.  17  gives  the  relation  between  the  output  current  and 
the  plate  or  anode  voltage.  When  using  the  tube  as  oscillator, 
we  operate  over  the  sloping  part  OA  of  the  characteristic.  The 
three  sets  of  curves  shown  are  for  different  values  of  the  filament 
current.  Fig.  18,  on  the  other  hand,  gives  the  relation  between 
the  anode  current  and  the  temperature  of  the  filament  or  the  fila- 
ment current.  But  the  sloping  part  of  this  characteristic  repre- 
sents a  temperature  of  the  filament  which  is  so  low  that  the  plate 
potential  is  sufficiently  high  to  draw  all  the  electrons  away  to  the 
plate  as  fast  as  they  are  emitted  from  the  filament.  The  condition 
which  may  be  characterized  as  temperature  saturation  is  repre- 
sented by  the  horizontal  poition  CD  of  the  curve,  and  obtains 
when  the  number  of  electrons  drawn  to  the  plate  is  less  than  the 
total  number  emitted.  The  part  CD  of  Fig.  18,  corresponds  to 
the  sloping  part  of  OA  of  Fig.  17;  hence,  for  a  given  d-c.  plate 
potential  it  is  necessary  that  the  temperature  of  the  filament  be  so 
high  that  we  operate  on  the  horizontal  part  of  the  plate  current, 
filament  current  characteristic.  If  this  is  not  the  case,  the  varia- 
tion in  output  current  with  the  variation  in  the  grid  potential  is 
too  small  to  produce  oscillations.  The  dependence  of  the  oscilla- 
tion current  and  the  plate  current  upon  the  filament  current  is 
indicated  in  Fig.  180.  If  the  filament  current  is  below  a  certain 
value  given  by  A,  the  tube  does  not  produce  sustained  oscillations. 
Filament  currents  below  this  value  correspond  to  the  saturation 
part  of  the  curve  giving  the  plate  current  as  a  function  of  the  plate 
potential.  If  the  filament  current  is  raised  beyond  the  value 
indicated  by  A,  the  tube  starts  oscillating  and  the  oscillation 
current  increases  until,  when  temperature  saturation  is  obtained, 
it  shows  no  further  increase  with  increase  in  filament  current. 
In  order  to  secure  the  best  operation,  therefore,  the  filament  cur- 
rent should  not  be  less  than  the  value  indicated  by  B.  On  the 

1  A  variety  of  experimental  curves  have  been  obtained  by  Heising  with  a 
standard  VT-2  type  of  tube  and  published  in  the  Journal  of  the  A.I.E.E., 
May,  1920. 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR        309 

other  hand,  the  filament  current  should  not  be  increased  much 
beyond  this  value  because  that  would  shorten  the  life  of  the  tube. 
If  the  tube  is  operated  with  a  resistance  Rs  in  the  grid  circuit, 
as  indicated  in  Fig.  169,  for  example,  the  value  of  the  oscillation 
current  obtained  depends  on  this  resistance,  in  the  manner  shown 
in  Fig.  181,  where  the  lowest  curve  represents  the  highest  leak 
resistance  Rs  in  the  grid  circuit.  The  oscillation  current  is  less 
for  the  higher  resistance,  but  the  horizontal  part  of  the  curve 


Filament  Current 
FIG.  180. 

starts  at  a  lower  filament  current.  On  the  other  hand,  if  the 
oscillation  current  be  plotted  as  a  function  of  the  filament  current 
for  various  values  of  the  plate  potential,  a  set  of  curves  is  obtained 
similar  to  that  shown  in  Fig.  181,  except  that  the  lowest  curve 
would  represent  the  case  for  the  lowest  plate  potential,  so  that 
although  the  output  can  be  increased  by  increasing  the  plate  bat- 
tery voltage,  the  horizontal  part  of  the  curve  is  reached  at  a  higher 
filament  current.  The  filament  current  at  which  the  bend  in  the 
curve  occurs  can  be  taken  to  represent  the  safe  temperature  of 
the  filament.  It  will  be  seen  then  that  the  safe  temperature 


310 


THERMIONIC  VACUUM  TUBE 


increases  with  increase  in  plate  potential  and  decreases  with 
increase  in  the  grid  leak  resistance.  By  making  use  of  these 
two  variables,  plate  potential  and  grid  leak  resistance,  a  com- 
promise can  be  effected  to  give  the  best  output  for  the  longest  life 
of  the  filament. 

The  relation  between  the  oscillation  current  and  plate  poten- 
tial is  shown  in  Fig.  182.1  The  tube  starts  oscillating  at  a  plate 
potential  depending  on  the  adjustments  of  the  circuit.  If  the 
plate  voltage  be  raised,  the  oscillation  current  increases  almost 


Filament  Current 
FIG.  181. 

linearly  with  it.  As  the  grid  leak  resistance  is  increased,  the  slope 
of  this  line  becomes  less  and  the  oscillation  current  for  given  plate 
potential  becomes  less.  The  value  of  leak  resistance  Rs  that  gives 
satisfactory  operation  usually  lies  in  the  neighborhood  of  5000  to 
10,000  ohms. 

When  a  grid  battery  is  used  to  maintain  the  grid  at  an  appro- 
priate negative  potential,  the  tube  behaves  differently  from  the 
manner  explained  above,  where  the  negative  grid  potential  was 
maintained  by  means  of  the  grid  leak  resistance  Rs.  For  example, 
with  a  battery  in  the  grid  circuit  the  oscillations  will  usually  not 
1 R.  A.  HEISING,  loc.  cit. 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR        311 


start  until  the  plate  voltage  is  raised  to  a  higher  value  than  that 
necessary  when  using  the  grid  leak  resistance  instead  of  a  battery. 


0.8 
0.7 


o. 


O.I 


ioo     eoo    300    400    500 

Plate  Voltage 
FIG.  182 


100      ZOO       300       400       500' 
Plate  Volts 

FIG.  183. 


If  the  plate  voltage  be  further  increased,  the  oscillation  current 
increases  almost  linearly  as  indicated  in  Fig.  183.     But  if  the  plate 


312 


THERMIONIC  VACUUM  TUBE 


voltage  be  again  reduced,  the  oscillations  will  persist  until  this 
voltage  reaches  a  value  which  is  quite  appreciably  lower  than 
that  necessary  to  start  the  oscillations. 

The  output  power  as  a  function  of  the  plate  battery  voltage 
can  be  represented  by  a  curve  like  that  shown  in  Fig.  184,  which 
shows  a  rather  rapid  increase  as  the  plate  voltage  is  raised.  To 
obtain  increase  in  output  power  by  increasing  the  plate  voltage, 
it  is,  of  course,  necessary  to  insure  that  the  filament  current  is 


100       ZOO       300       400      500 
Plate  Volts 

FIG.  184. 

high  enough  to  prevent  the  space  current  from  becoming  sat- 
urated. Also,  the  power  delivered  to  the  oscillation  circuit  depends 
on  the  resistance  of  this  circuit  and  the  resistance  of  the  tube.  The 
latter  depends  on  the  d-c.  plate  potential  so  that  in  general  an 
increase  in  the  plate  battery  voltage  would  necessitate  a  read- 
justment of  the  capacity  and  inductance  in  the  oscillation  circuit 
to  give  the  maximum  output  power. 

96.  Range  of  Frequency  Obtainable  with  the  Vacuum  Tube 
Oscillator.  Circuits  for  Extreme  Frequencies.  The  vacuum 
tube  has  been  used  to  give  oscillations  having  a  frequency  ranging 


VACUUM  TUBE  AS  AN  OSCILLATION  GENERATOR        313 


from  a  fraction  of  a  cycle  per  second  to  many  millions  of  cycles  per 
second.  For  low  frequencies,  the  frequency  is  determined  almost 
entirely  by  the  inductance  and  capacity  in  the  oscillation  circuit, 
and  the  only  limitation  to  this  end  of  the  scale  is  the  size  cf  the 
inductances  and  capacities.  For  very  high  frequencies,  the  fre- 
quency of  the  oscillation  is  determined  mainly  by  the  electrostatic 
capacity  between  the  electrodes  of  the  tube  and  by  the  inductances 
and  capacities  of  the  wires  connecting  the  electrodes.  The  upper 
limit  to  the  frequency  obtainable  depends  mainly  on  the  intra- 
electrode  capacities. 


FIG.  185 

When  very  low  frequencies  are  desired,  it  is  best  to  use  a  Hart- 
ley circuit,  in  which  the  two  coils  LI  and  Z/2  of  Fig.  159  take  a  form 
of  an  iron  core  transformer  such  as  is  shown  in  Fig.  185.  By 
means  of  such  a  circuit  it  has  been  possible  to  obtain  frequencies 
as  small  as  a  fraction  of  a  cycle  per  second. 

When  it  is  desired  to  obtain  exceptionally  high  frequencies, 
the  inductances  in  the  oscillation  circuit  can  be  reduced  to  the 
utmost  extent,  until  they  take  the  form  of  short  straight  wires 
connecting  the  electrodes.  The  capacity  between  grid  and  plate 
forms  the  capacity  of  the  oscillation  circuit.  A  circuit  which 
has  been  used,  for  example,  by  W.  C.  White,1  to  obtain  a  fre- 

1  General  Electric  Review,  Vol.  19,  771,  1916? 


314 


THERMIONIC  VACUUM  TUBE 


quency  of  fifty  million  cycles  per  second  is  shown  in  Fig.  186. 
The  grid  inductance  is  furnished  by  the  connecting  wire  GAP  and 
the  plate  inductance  by  the  connecting  wire  PDB.  The  plate 
current  is  supplied  by  the  battery  Ej,  through  the  choke  coil  LI. 
Ci  represents  a  by-pass  for  the  high  frequency  and  is  so  large  that 
it  does  not  affect  the  oscillation  frequency.  W  represents  a  long 
pair  of  parallel  wires  connected  to  the  system  through  the  small 
capacities  €2  and  €3.  By  suitably  adjusting  the  bridge  H,  stand- 


FIG.  186. 

ing  waves  can  be  obtained.  In  White's  experiments  these  waves 
were  about  6  meters  long.  This  circuit  represents  a  very  simple 
means  of  demonstrating  standing  waves.  The  vacuum  tube  is 
much  superior  to  the  induction  coil  frequently  used  in  laboratories 
for  this  demonstration  experiment.  By  using  tubes  that  are 
specially  designed  to  have  low  electrostatic  capacities  between  its 
electrodes,  it  is  possible  to  obtain  waves  of  a  few  feet  in  length.  . 


CHAPTER  IX 

MODULATION  AND  DETECTION  OF  CURRENTS  WITH  THE 
VACUUM  TUBE 

97.  Elementary  Theory  of  Modulation  and  Detection.    In  the 

applications  of  the  vacuum  tube  considered  so  far,  it  is  desirable 
that  the  characteristic  of  the  plate  circuit  be  as  straight  as  possible. 
For  example,  when  using  the  device  as  an  amplifier,  it  was  explained 
in  Chapter  VII  that  the  external  impedance  in  the  plate  circuit  is 
usually  made  so  large  that  the  current  voltage  characteristic  of 
the  plate  circuit  is  sufficiently  straightened  out  to  enable  us  to 
neglect  quantities  of  higher  order  than  the  first.  When  using  the 
tube  as  an  oscillation  generator,  it  is  also  desirable  to  have  a  linear 
characteristic  because  the  curvature  introduces  harmonics  which 
result  in  a  waste  of  power. 

In  the  following  we  shall  consider  those  applications  of  the 
vacuum  tube  which  depend  directly  on  the  curvature  of  the 
characteristic.  The  two  most  important  of  these  applications 
are  the  use  of  the  tube  to  modulate  high  frequency  currents  for 
purposes  of  signaling  and  the  detection  of  high  frequency  currents. 

When  considering  the  second  order  quantities  that  enter  into 
the  characteristic  of  the  device,  it  is  generally  not  possible  to 
express  the  characteristic  by  a  simple  equation,  but  we  can  still 
apply  the  equation  derived  in  Section  22,  Chapter  III,  which 
holds  generally  for  three  electrode  devices.  Neglecting  the 
small  quantity  we  can  write  this  equation  in  the  form 


In  general  the  function  /  is  not  linear  and,  therefore,  if  a  sinusoidal 
voltage  be  impressed  upon  the  input  of  the  tube,  the  output  wave 
will  be  distorted  in  the  manner  explained  in  Section  57.  For 
such  a  condition  we  can  express  the  varying  current  in  the  output 

315 


316  THERMIONIC  VACUUM  TUBE 

as  a  function  of  the  sinusoidal  voltage  impressed  upon  the  input 
by  a  simple  power  series, 


(2) 


where  J  represents  the  varying  current  and  will  in  general  have 
the  form  of  a  lopsided  wave,  and  will,  therefore,  comprise  currents 
of  different  frequencies  and  a  direct-current  component.  This 
series,  it  has  been  found,  usually  converges  so  rapidly  that  we  can 
neglect  all  quantities  of  higher  order  than  the  second.  Experi- 
mental proof  of  this  will  be  given  later  on.  The  first  term  of 
equation  (2)  represents  a  current  having  the  same  frequency 
as  that  of  the  input  voltage  e.  The  second  term  is  the  one  which 
gives  rise  to  modulation  and  detection  effects. 

To  evaluate  the  coefficients  a\  and  02,  we  can  proceed  in  the 
manner  given  by  J.  R.  Carson.1  Carson  has  considered  two 
cases,  namely,  when  the  output  circuit  contains  a  pure  non- 
inductive  resistance  and  secondly  when  the  output  circuit  contains 
a  general  impedance.  In  order  to  derive  an  expression  for  the 
coefficient  0,2  in  terms  of  the  parameters  of  the  tube,  we  khall 
discuss  only  the  first  case,  namely,  in  which  the  plate  circuit 
contains  only  a  pure  resistance. 

The  quantities  to  be  considered  in  the  circuit  can  be  expressed 
as  follows: 


E0=Ec+e 


(3) 


where  I&,  Eb  and  Ec  represent  the  d-c.  values  of  plate  current  and 
potential,  and  grid  potential,  and  Ip,  EF  and  E0  are  the  quantities 
that  obtain  as  a  result  of  the  variations  J,  v  and  e  superimposed  on 
the  d-c.  values.  Substitution  of  equation  (3)  into  (2)  gives: 


......     (4) 

where  PI,  P2,  etc.,  are  given  by 


P  -L 

-n\(3E»JEp=Eb-         •    '    .....     (5) 

1  J.  R.  CARSON,  Proceedings  I.R.E.,  Vol.  7,  p.  187,  1919. 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    317 

The  physical  significance  of  the  differential  parameters  Pn  become 
apparent  when  they  are  evaluated  with  the  help  of  the  character- 
istic equation  (1).  Thus, 


P  = 


2r2 


(6) 


where  rp  is  the  plate  resistance  of  the  tube  and  r'P  is  the  variation 
in  the  plate  resistance  due  to  the  curvature.  These  equations  now 
enable  us  to  evaluate  the  coefficients  ai  and  0,2  of  equation  (2). 
To  do  this  it  should  be  noted  that  the  variation  v  in  plate  voltage 
is  equal  and  opposite  to  the  voltage  drop  established  in  the  external 
resistance  TQ  due  to  the  varying  current  /  in  the  output.  Hence, 
substituting  v=  —roJ  into  equation  (4),  we  get: 

This  equation  now  gives  the  varying  current  /  in  terms  of  the 
input  voltage  e  and  the  parameters  of  the  tube  and  circuit.  To 
express  J  as  an  explicit  function  of  the  input,  we  can  substitute 
the  series  for  J  given  by  equation  (2)  into  equation  (7)  and  equate 
coefficients  of  like  powers  of  e.  When  this  is  done  the  expression 
for  the  varying  current  J  becomes, 

UP  1    /|2r'  r  P*> 

T       fj,e        i  v  r  prpe 


If  the  characteristic  is  linear,  the  plate  resistance  rp  is  constant 
and,  therefore,  its  derivative  r'p  is  zero.  This  makes  the  second 
term  of  the  above  equation  zero.  Hence,  replacing  the  varying 
values  J  and  e  by  the  R.M.S.  values  ip  ^nd  effj  equation  (8)  reduces 
to  equation  (22)  given  in  Chapter  VII. 

The  second  term  of  equation  (8)  represents  the  property  of 
the  tube  that  enables  it  to  act  as  a  modulator  and  detector. 
The  value  of  the  coefficient  given  by  the  second  term  in  equation 
(8)  will  be  helpful  in  the  interpretation  of  the  equations  that 
follow.  For  the  present  we  shall  use  equation  (8)  in  the  simple 
form 

.     (9) 


318 


THERMIONIC  VACUUM  TUBE 


to  explain  how  the  second  term  is  instrumental  in  producing  mod- 
ulation and  detection. 

98.  Modulation.  Suppose  that  a  tube  be  inserted  in  a  cir- 
cuit such  as  that  shown  in  Fig.  187.  Let  high  frequency  currents 
be  impressed  at  H.  F.  and  low  frequency  currents,  lying  within 
the  audible  range,  at  L.  F.  The  total  input  voltage  on  the  tube 
is  then, 

«=«i  sin  pt+e2  sin  qt,       .....     (10) 


where  £-  and  £•  represent  the  high  and  the  low  frequencies 
ZTT  2iir 

respectively.     In  order  to  obtain  the   output  current,  we  have 
to  substitute  this  expression  for  e  in  equation  (9).     When  using  the 


mm 

\          L.F. 


FIG.  187. 


tube  as  a  modulator,  we  are  interested  only  in  currents  having 
frequencies  lying  within  the  range  ^— - .     Hence,   substituting 

(10)  into  (9),  evaluating  and  dropping  all  terms  having  frequencies 
lying  outside  of  this  range,  we  obtain, 

J  =  a\e\  sin  pt-\-2a,2eie2  sin  pt  sin  qt.     .     .     .     (11) 

This  expression  represents  a  wave  of  varying  amplitude  as  shown 
in  Fig.  188,  the  amplitude  of  the  high  frequency  carrier1  wave 

1  The  word  "  carrier  "  is  here  used  as  a  general  term  to  indicate  the  high  . 
frequency  wave,  which  is  modulated  by  the  signaling  wave.     It  has  also  a 
more  specific  meaning  in  which  it  refers  to  the  transmission  of  high  fre- 
quency currents  over  wires. 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    319 

varying  in  accordance  with  the  audio  frequency  wave   impressed 
on  the  input  of  the  tube. 

We  can,  for  purposes  of  explanation,  write  equation  (11)  in 
the  form, 

J  =  Asmpt(l+Bsmqt).         ....     (12) 


FIG.  188. 


b 


A 
FIG.  189. 


The  way  in  which  a  wave  of  the  type  shown  in  Fig.  188  is  pro- 
duced by  the  vacuum  tube,  becomes  apparent  when  we  consider 
the  characteristic.  For  example,  Fig.  189  shows  the  plate  cur- 
rent grid  potential  characteristic.  Suppose  a  constant  potential 


320 


THERMIONIC  VACUUM  TUBE 


EC  be  applied  to  the  grid  so  that  the  normal  plate  current  is  rep- 
resented by  the  ordinate  AO.  Now  let  a  high  frequency  voltage 
of  amplitude  Oa  be  superimposed  on  this  constant  grid  potential. 
The  output  of  current  wave  will  then  have  the  amplitude  given  by 
ab.  If  the  grid  potential  be  increased  to  the  value  BC  and  a  high 
frequency  voltage  of  the  same  amplitude  as  before  be  impressed 
on  the  input,  the  output  current  wave  will  have  an  amplitude 
a"b",  and  this  is  smaller  than  before.  If,  on  the  other  hand,  the 
grid  potential  be  reduced  to  the  value  DC,  the  amplitude  of  the 


FIG.  190. 

output  current  wave  for  the  same  amplitude  of  input  becomes 
greater  and  is  represented  by  a'b'.  If  now,  we  impress  on  the 
input  not  only  a  high  frequency  voltage  of  constant  amplitude 
Oa,  but  also  at  the  same  time  a  low  frequency  having  an  amplitude 
equal  to  say  AB,  then  the  amplitude  of  the  output  current  wave 
will  alternately  increase  and  decrease  at  a  frequency  equal  to  that 
of  the  low  frequency  wave  impressed  on  the  input  and  the  result 
is  an  output  current  wave  of  the  shape  shown  in  Fig.  188.  Fig. 
190  represents  the  input  and  output  waves.  The  input  wave  is  a 
high  frequency  of  constant  amplitude  superimposed  on  a  low  fre- 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    321 

quency,  while  the  output  wave  is  a  high  frequency  of  varying 
amplitude  superimposed  on  a  low  frequency.  If  the  output  circuit 
(Fig.  187)  be  tuned  to  the  high  frequency,  the  low  frequency  cur- 
rent variations  are  filtered  out,  thus  resulting  in  the  wave  shown  in 
Fig.  188. 

If  the  low  frequency  voltage  impressed  on  the  input  circuit 
has  such  a  value  that  the  maximum  negative  potential  of  the  grid 
becomes  equal  to  CF  (Fig.  189),  the  current  is  reduced  to  zero  and 
the  modulated  output  wave  then  takes  the  form  shown  in  Fig.  191. 
The  wave  can  then  be  said  to  be  completely  modulated.  When 
this  happens  the  coefficient  B  in  equation  (12)  is  unity  and  the 
maximum  amplitude  of  the  high  frequency  wave  when  the  grid 
has  its  minimum  negative  potential  is  then  2 A .  In  some  measure- 
ments it  is  very  important  to  insure  that  the  wave  is  completely 


FIG.  191. 

modulated  as  will  become  evident  later  on  when  we  come  to  con- 
sider measurements  on  the  detecting  efficiency  of  tubes. 

The  second  term  in  equation  (11)  gives  a  measure  of  the  extent 
to  which  a  wave  is  modulated.  The  coefficient  «2  is  given  by 
equation  (8),  namely, 


(13) 


The  amplitude  of  the  modulated  wave  is,  therefore,  proportional 
to 


€162 


(14) 


It  is,  therefore,  proportional  to  the  product  of  the  amplitudes  of 
the  audio  and  the  radio  input  voltages  and  to  the  curvature  r'p 
of  the  characteristic.  The  modulated  output  power  is  also  pro- 


portional to  -.  —  -~-  -rg,  and  this  is  a  maximum  when  ro  is  equal  to 


322  THERMIONIC  VACUUM  TUBE 

$rp,  a  result  which  was   stated   by  Carson.1     If  we   put  ro  =  nrp 
expression  (14)  may  be  written: 


which  shows  that  the  value  of  the  device  as  a  modulator  depends 
on  the  ratio  of  n  to  rp.  This  quantity  which  is  the  mutual  con- 
ductance of  the  tube  has  also  been  found  to  be  a  measure  of  the 
figure  of  merit  of  the  tube  as  amplifier  and  as  oscillation  generator. 
99.  Modulation  Systems.  The  results  derived  above  can  be 
interpreted  by  stating  that  a  device  will  operate  as  a  modulator 
if  it  has  a  varying  resistance  characteristic;  the  resistance  to  the 
radio  frequency  currents  is  varied  in  accordance  with  audio  fre- 
quency currents.  There  are,  therefore,  two  main  systems  whereby 
modulated  currents  can  be  transmitted  over  a  line  or  from  an 
antenna.  The  first  is  exemplified  in  Fig.  187.  Radio  frequency 
and  audio  frequency  voltages  are  impressed  on  the  grid  and  the 
resulting  modulated  current  in  the  output  of  the  tube  is  trans- 
mitted over  a  line  or  antenna  of  constant  impedance.  The 


antenna  must  then  be  tuned  to  a  frequency  range    75—,  where     - 

—  ••  £ir 

is  the  carrier  or  radio  frequency  and  ~  the  audio  frequency. 

£ir 

In  telephony  ^-  covers  a  range  of  from  about  100  to  2000  or  3000 

2ir 

cycles  per  second.  The  antenna  must;  therefore,  be  tuned  so 
that  it  has  approximately  the  same  impedance  for  frequencies 
covering  a  range  of  about  2000  cycles.  This  is  also  a  condition 
for  ordinary  wire  telephony  which  requires  that  the  telephone  line 
should  be  capable  of  transmitting  this  whole  range  of  frequencies 
with  about  equal  facility.  The  only  difference  is  that  in  ordinary 
wire  telephony  the  frequencies  cover  a  range  up  to  about  2000, 
whereas  in  carrier  or  radio  telephony  the  frequencies  cover  the 
same  range  but  their  actual  values  are  in  the  neighborhood  of  the 
carrier  frequency. 

The  other  system  consists  in  impressing  a  high  frequency 
directly  on  the  antenna  or  line  and  then  varying  the  resistance 
of  the  antenna  in  accordance  with  audio  frequency.  Such  a 
system  is  shown  schematically  in  Fig.  192,  which  shows  the  modu- 

:Loc.  cit. 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    323 

lator  M  in  shunt  with  the  antenna  inductance,  that  is,  the  antenna 
inductance  is  shunted  by  the  plate  resistance  of  the  modulator 
tube.  The  amount  of  current  in  the  antenna,  which  is  supplied 
by  the  high  frequency  oscillation  generator,  will,  therefore,  depend 
on  the  resistance  of  the  tube  M.  This  resistance  is  varied  in 
accordance  with  the  speech  voltages  impressed  on  its  input  in  the 
manner  explained  with  reference  to  Fig.  189. 

A  modification  given  by  R.  A.  Heising  l  is  shown  in  Fig.  193. 
The  oscillation  circuit  shown  here  is  the  same  as  that  given  in  Fig. 
179,  the  capacity  of  the  antenna  forming  the  capacity  €2  of  Fig. 


FIG.  192. 

179.  The  oscillator  and  modulator  are  both  supplied  by  a  battery- 
through  a  low  frequency  choke  coil  which  insures  that  they  are 
both  supplied  with  constant  direct  current.  The  speech  or  audio 
frequency  voltage  is  impressed  on  the  grid  of  the  modulator  by 
means  of  the  transmitter  through  the  transformer  as  indicated. 
Between  the  plates  of  the  modulator  and  oscillator  is  a  high  fre- 
quency choke.  If  the  telephone  transmitter  is  not  actuated,  the 
oscillator  tube  supplies  high  frequency  currents  of  constant  ampli- 
tude to  the  antenna.  If  now  an  audio  frequency  voltage  be 
impressed  on  the  grid  of  the  modulator,  audio  frequency  currents 
are  established  in  the  output  circuit  of  this  tube  and  consequently 
1  See  CRAFT  and  COLPITTS,  Proceedings,  A.I.E.E.,  Vol.  38,  p.  360,  1919. 


324 


THERMIONIC  VACUUM  TUBE 


the  potential  of  the  plate  of  the  oscillator  varies  in  accordance 
with  the  low  frequency,  thus  producing  low  frequency  variations 
in  the  amplitude  of  the  high  frequency  oscillations  obtained  from 
the  oscillator  and  impressed  on  the  antenna.  The  coupling  is 
adjusted  by  sliding  the  contact  Q  as  explained  in  connection  with 
Fig.  179  in  Chapter  VIII. 

A  number  of  modulating  and  transmitting  circuits  have  been 
suggested.  The  circuit  shown  in  Fig.  194  J  is  another  illustration 
of  the  application  of  the  principles  given  in  the  foregoing.  This 
circuit  is  so  arranged  that  the  high  frequency  is  impressed  at  H.  F. 
in  such  a  way  that  the  grids  of  both  tubes  are  in  phase,  The  high 


Low  Frequency 
Choke 


High  Frequency 
Choke 


FIG.  193. 


frequency  currents  in  the  output  coils,  therefore,  flow  in  opposite 
directions  and  the  output  in  the  secondary  of  the  transformer  To 
is  zero.  But  if  the  audio  frequency  voltage  is  impressed  as  indi- 
cated at  L.  F.,  the  grid  of  the  one  tube  becomes  positive  wheYi 
the  other  becomes  negative  so  that  the  resistance  of  the  one  tube 
is  reduced  while  that  of  the  other  is  increased.  This  causes  an 
increase  in  the  amplitude  of  the  high  frequency  currents  flowing 
through  the  one  tube  and  a  decrease  in  the  amplitude  of  those 
flowing  through  the  other  tube.  In  this  way,  therefore,  energy 
is  radiated  only  during  the  time  that  the  tube  is  actuated  by  the 

1  British  Patent  130219,  1918. 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    325 


speech  voltage.    What  is  transmitted  then  is  only  the  waves 
given  by  the  second  term  of  equation  (11). 

100.  Detection.  The  mechanism  of  detection  is  identical 
with  that  of  modulation  and  is  due  to  the  same  cause,  namely, 
the  curvature  of  the  characteristic.  In  general,  therefore,  the 
equations  derived  above  are  applicable  also  to  the  problem  of 
radio  detection  with  the  thermionic  tube.  The  only  difference 
is  that  in  this  case  we  are  concerned  with  a  different  range  of 
frequencies.  While  in  the  case  of  the  modulator,  the  output  is 
tuned  to  radio  frequencies,  in  the  detector  the  output  is  tuned 
to  audio  frequencies  because  the  problem  of  detection  involves 


FIG.  194. 


transforming  high  frequency  into  low  frequency  currents  so  that 
they  can  become  audible.  We  can,  therefore,  use  equation  (9) 
to  determine  the  low  frequency  output  of  a  detector.  In  this 
case,  however,  we  are  not  concerned  with  the  first  term  at  all. 
For  example,  if  a  radio  frequency  e  sin  pt  be  impressed  upon  the 
input  of  a  detector,  the  output  current  is  given  by: 


=  aie  sn 


--  -r  cos  2  pt. 


(16) 


The  first  term  is  simply  an  inaudible  high  frequency.  It  need, 
therefore,  not  be  considered  and  we  can  write  instead  of  equation 
(9)  the  equation  for  the  instantaneous  detecting  current  Id  as 


la  =  ae2  sin2 


pt} 


(17) 


326  THERMIONIC  VACUUM  TUBE 

where  a  is  written  for  02.  We  shall  refer  to  a  as  the  detection 
coefficient.  Its  value  in  terms  of  the  parameters  of  the  circuit 
is  given  by  equation  (13). 

Equation  (16)  contains  only  high  frequency  components  and  a 
d-c.  component.  The  d-c.  component  of  equation  (16)  makes 
possible  the  detection  of  high  frequency  incoming  currents  im- 
pressed on  the  input  of  the  detector,  if  the  output  of  the  detector 
contains  a  d-c.  current  measuring  instrument  which  is  sensitive 
enough  to  indicate  a  change  in  the  plate  current  given  by  the  sec- 
ond term  of  equation  (16).  When  a  telephone  receiver  is  used  in 
the  output  continuous  incoming  waves  of  constant  amplitude 
cannot  be  detected,  because  equation  (16)  does  not  contain  an 
audio  frequency  term.  The  incoming  waves  must  either  be  mod- 
ulated high  frequency  waves  or  if  they  are  continuous  waves 
6f  constant  amplitude,  the  heterodyne  method  must  be  used  to 
detect  them  (see  Section  109).  If  a  modulated  high  frequency 
wave  such  as  that  given  by  equation  (12)  be  impressed  on  the 
detector,  the  instantaneous  value  of  the  detecting  current  is  given 

by 

Id  =  a\A  sin  pt  (l+B  sin  qt)]2.     ..  ' .    .     .     (18) 

In  evaluating  this  expression,  all  terms  containing  frequencies 
that  lie  outside  the  audible  range  can  be  neglected.  This  gives: 

Id  =  aA2  Bsmqt-^-j — cos  2  qt.   ....     .     (19) 

Now  q  in  equation  (12)  represents  the  low  frequency  component 
of  the  modulated  wave.  It  is  seen,  therefore,  that  in  view  of  the 
curvature  of  the  characteristic  cf  the  tube  the  output  current 
contains  a  term  of  the  same  frequency  as  the  audio  frequency  with 
which  the  carrier  wave  was  modulated.  It  also  contains  a  term 
having  twice  the  audio  frequency.  This  term  is,  however,  usually 
so  small  as  not  to  cause  any  appreciable  distortion  of  the  wave  in 
the  output  of  the  detector. 

In  deriving  these  expressions  it  is  assumed  that  the  grid  does 
not  take  appreciable  current.  The  circuit  in  which  the  detector 
can  be  used  to  comply  with  the  above  equations  is  shown  in  Fig. 
195.  The  input  circuit  LC  is  tuned  to  the  frequency  of  the  incom- 
ing oscillations  and  the  grid  is  kept  negative  with  respect  to  the 
filament  by  means  of  the  battery  Eg.  The  condenser  Ci  serves  as 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE  327 

a  by-pass  to  the  high  frequency  currents  in  the  output  circuit,  the 
audio  frequency  component  of  the  output  passing  through  the 
telephone  receiver. 

A  receiving  circuit  that  is  commonly  used  and  in  which  the 
battery  Eg  is  replaced  by  a  condenser,  will  be  discussed  later  on. 
(Section  103.) 

It  will  be  apparent  that  the  reason  why  equation  (19)  con- 
tains a  term  having  the  same  frequency  ~  that  is  used  to  modu- 

2ir  . 

late  the  wave  at  the  transmitting  station,  is  because  the  incoming 
wave,  which  is  given  by  equation  (12),  contains  both  the  terms 
A  sin  pt  and  AB  sin  pt  sin  qt.  If  the  incoming  wave  were  of  the 


FIG.  195. 

form  C  sin  pt  sin  qt,  simple  trigonometry  will  show  that  the  only 
audio  component  of  the  current  in  the  output  of  the  detector  is 
one  which  has  double  the  modulating  frequency,  the  audio 
detecting  current  being  given  by: 


(20) 


which  on  evaluating  and  dropping  inaudible  terms  becomes: 


(21) 


It  follows  therefore  that  in  order  to  obtain  the  modulating 
frequency  ~,  the  waves  impressed  on  the  input  of  the  detector 


328  THERMIONIC  VACUUM  TUBE 

must  be  made  to  include  a  wave  of  the  desired  strength  having 
the  frequency  —-. 

4TT 

101.  Root  Mean  Square  Values  of  Detecting  and  Modulated 
Currents.  The  above  equations  give  the  instantaneous  values 
of  the  currents  or  voltages  considered.  The  R.M.S.  values  can 
readily  be  obtained.  Thus,  the  R.M.S.  value  id  of  the  detecting 
current,  the  instantaneous  value  of  which  is  given  by  equation 
(19)  is: 

(22) 


If  we  neglect  the  small  double  frequency  quantity  given  by  the 
second  term  in  the  parenthesis,  id  reduces  to  the  common  form 


The  R.M.S.  value  of  the  modulated  input  voltage  as  given  by 
equation  (12)  can  be  obtained  by  putting  p  =  nq,  since  p  is  large 

compared  with  q.      (^-  covers  frequencies    ranging  to    2000   or 

V 

3000  cycles  per  second,  while  J-  is  generally  of  the  order  of  sev- 

Zir 

eral  hundred  thousand  cycles  per  second).  The  R.M.S.  of  the 
modulated  wave  which  can  be  taken  as  the  effective  input  volt- 
age ee  on  the  grid  of  the  detector,  then  becomes  : 


(24) 


and  involves  B  which  is  a  measure  of  the  extent  to  which  the 
wave  is  modulated.  If  the  wave  is  completely  modulated  B=l, 
as  was  explained  in  Section  93.  In  this  case,  remembering  that 
the  peak  value  of  the  high  frequency  is  2A  we  find  that  the  ratio 

of  the  R.M.S.  to  the  peak  value  is  —7=  instead  of  V2  as  in  un- 

Vz 

damped  waves. 

102.  Relation  between  Detection  Coefficient  and  the  Operating 
Plate  and  Grid  Voltages.  The  detection  coefficient  a  depends 
on  the  values  of  the  d-c.  plate  and  grid  voltages  so  that  in  deter- 
mining the  value  of  a  tube  as  a  detector,  this  relationship  must  be 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    329 


taken  into  account.  If  the  detecting  current  id  be  measured  as  a 
function  of  the  effective  voltage  Ey=  (— +Eg+e}  it  will  be  found 

that  as  this  voltage  is  increased  by  increasing  either  Ep  or  Ea,  the 
detecting  current  at  first  increases,  reaches  a  maximum,  and  then 
decreases.  It  is  assumed  that  the  grid  is  at  all  times  negative 
with  respect  to  the  negative  end  of  the  filament.  Now  the 
detection  coefficient  a  is  given  by  the  second  derivative  of  the  char- 
acteristic, and  is  a  measure  of  the  detecting  current,  that  is,  the 
audio  frequency  component  in  the  output.  The  maximum  of 


FIG.  196. 

detecting  current  such  as  shown  in  Fig.  196  is  due  to  the  potential 
drop  in  the  filament  due  to  the  heating  current.  It  can  be  ac- 
counted for  if  we  take  regard  of  the  voltage  drop  in  the  filament  in 
giving  an  expression  for  the  current  as  a  function  of  the  plate  or 
grid  voltage.  It  was  shown  in  Section  28  that  if  this  be  con- 
sidered, the  characteristic  of  the  tube  can  be  expressed  by  means 
of  two  equations,  one  which  holds  for  values  of  the  applied  plate 
potential  less  than  the  potential  drop  in  the  filament  and  the  other 
for  larger  values  of  the  plate  potential.  These  two  equations  are 
given  as  equations  (17)  and  (19)  of  Chapter  IV.  They  were 
derived  for  the  case  of  a  simple  valve  containing  only  anode  and 


330  THERMIONIC  VACUUM  TUBE 

cathode.     But  we  can,  to  a  first  approximation,  apply  the  con- 
siderations given  there  to  the  three  electrode  device  if  we  replace 

,  /Fi  V 

the  plate  potential  by  the  expression  Ey=  (  —  '+Ea-\-  e  )  so  that  we 
can  write  the  characteristic  equations  in  the  form 


orEy^Ef,     ........     (25) 

(Ey-E,)5/*}forE^E,.     .     .     .     (26) 

where  Ef  is  the  voltage  drop  in  the  filament. 

These  two  equations  can  be  represented  by  a  continuous  curve 
closely  approximating  a  parabola.  The  detecting  current,  or 
the  second  derivative  of  equations  (25)  and  (26)  when  plotted 
as  a  function  of  the  effective  voltage  on  the  other  hand,  shows  a 
distinct  maximum,  which  occurs  at  a  value  of  the  effective  voltage 
Ey  equal  to  the  voltage  drop  in  the  filament.  The  simple  rule, 
therefore,  to  obtain  the  best  results  when  using  the  tube  in  the 
circuit  shown  in  Fig.  195  is  to  make 


(27) 


Fig.  197  shows  an  experimental  curve  in  which  the  detecting 
current  is  plotted  as  a  function  of  the  plate  potential  Ep,  the 
grid  potential  remaining  constant.  For  n  =  l2,  €=—0.5,  #/=2.5 
volts  and  Eg  =  Q,  the  maximum  according  to  equation  (27)  occurs 
at  a  plate  potential  of  about  36  volts. 

The  condition  given  by  equation  (27)  states  that  the  potential 
difference  between  a  plane  coincident  with  that  of  the  grid  and 
the  positive  end  of  the  filament  is  zero.  This  condition  holds  gen- 
erally even  when  the  grid  is  connected  to  the  positive  end  of  the 
filament  instead  of  to  the  negative.  If  it  is  connected  to  the  pos- 
itive end  the  condition  for  maximum  detecting  current  is  Ey  =  Q. 
This  has  also  been  verified  experimentally.  The  condition 
Ey  =  0  when  the  grid  is  connected  to  the  positive  end  of  the  filament 
does,  of  course,  not  mean  that  the  space  current  is  zero  because 
since  Ey  is  positive  when  reckoned  from  all  points  on  the  filament 
other  than  the  extreme  positive  end. 

When  using  the  tube  in  the  simple  circuit  shown  in  Fig.  195 
it  is  necessary  to  make  sure  that  electrons  do  not  flow  to  the  grid. 
This  is  usually  secured  by  putting  in  the  negative  grid  battery  Eg. 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    331 


In  practice,  especially  when  receiving  weak  signals,  it  is  usually 
not  necessary  to  insert  this  battery  because  the  potential  varia- 
tions impressed  on  the  grid  seldom  exceed  a  small  fraction  of  a  volt, 
and,  under  these  conditions,  the  current  flowing  to  the  grid  is 
usually  negligibly  small.  There  is,  however,  a  factor  which  must 
be  considered,  namely  the  contact  potential  difference  between 


140x10  r 


120 


100 


60 


40 


40 
Plate  Voltage 

FIG.  197. 

the  filament  and  the  system  constituting  grid  and  plate.  The 
quantity  e  in  the  characteristic  equation  gives  a  measure  of  this 
effect.  If  the  filament,  grid  and  plate  are  of  the  same  material  e 
will  usually  be  practically  zero,  but  if  the  filament  is,  for  example, 
of  a  different  material  e  may  be  either  positive  or  negative,  but  it 
seldom  exceeds  the  value  of  about  1  volt.  If  it  is  positive  it  means 
that  the  grid  is  intrinsically  positive  with  respect  to  the  filament 
and,  therefore,  to  secure  best  operation  it  is  necessary  to  insert 


332 


THERMIONIC  VACUUM  TUBE 


a  grid  battery  to  maintain  the  resultant  potential  of  the  grid 
negative  with  respect  to  the  negative  end  of  the  filament.  In 
the  case  of  tubes  containing  oxide  coated  filaments,  e  is  usually 
negative.  In  such  a  case,  therefore,  the  grid  battery  can  be  dis- 
pensed with  altogether.  The  quantity  e  will  differ  from  zero 
whenever  the  electron  affinity  of  the  filament  is  different  from  that 
of  the  grid,  the  contact  potential  difference  between  the  two  being 
equal  to  the  difference  between  their  electron  affinities  expressed 
in  volts  (see  Chapter  III). 

103.  Detection  with  Blocking  Condenser  in  Grid  Circuit.  The 
method  of  detection  discussed  above  and  which  can  be  carried  out 
in  practice  with  a  circuit  like  that  shown  in  Fig.  195  is  perhaps 
not  used  as  commonly  as  another  type  of  circuit  which  is  shown 


FIG.  198. 

in  Fig.  198.  The  difference  between  these  two  circuits  is  that 
Fig.  198  contains  in  the  grid  circuit  a  condenser  Cs  shunted  by  a 
high  resistance  leak  Rs.  The  mechanism  of  detection  with 
such  a  circuit  is  different  from  that  in  which  the  blocking  con- 
denser is  omitted.  In  the  latter  case  the  best  results  are  obtained 
when  the  grid  is  maintained  at  a  sufficiently  high  negative  poten- 
tial to  prevent  any  convection  current  from  flowing  between  fila- 
ment and  grid,  the  detection  depending  only  on  the  curvature  of 
the  plate-current  characteristic.  When  the  blocking  condenser  is 
used  the  detection  depends  on  the  curvature  of  the  grid-current 
characteristic,  the  potentials  of  the  elements  being  so  propor- 
tioned that  convection  current  does  flow  from  filament  to  grid. 

In  order  to  explain  how  the  tube  detects  with  a  condenser  in 
the  grid  circuit,  let  us  first  indicate  briefly  how  the  tube  operates 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    333 

without  the  blocking  condenser.  Fig.  199  shows  the  processes 
involved  in  this  case.  Modulated  high  frequency  potential  varia- 
tions are  impressed  on  the  grid.  On  account  of  the  curvature  of 
the  characteristic  the  high  frequency  current  variations  in  the  plate 
circuit  can  be  represented  by  a  lopsided  wave  curve.  This  effect 
was  explained  in  Section  57.  Such  a  lopsided  wave  gives  rise  to 


Grid  Po+eh-Hal 


PI  are  Current 


Audio  Current 
in  Telephone 


FIG.  199. 


the  audio  frequency  component  as  shown  in  the  bottom  curve  of 
Fig.  199. 

When  the  blocking  condenser  is  used  in  the  grid  circuit  the 
operation  of  the  tube  as  a  detector  is  as  follows:  Suppose  the 
incoming  oscillations  are  again  high  frequency  currents  modulated 
by  a  low  frequency  as  shown  by  the  uppermost  curve  of  Fig.  200. 
Suppose  for  the  present  that  the  resistance  Rs  is  omitted.  When 
the  grid  potential  becomes  positive  with  respect  to  that  of  the 
filament,  electrons  are  attracted  to  the  grid.  During  the  next 
half  cycle  when  the  grid  potential  becomes  negative  the  electrons 


334 


THERMIONIC  VACUUM  TUBE 


cannot  escape  from  the  grid  because  they  are  trapped  on  the 
insulated  part  of  the  circuit  comprising  the  grid  and  the  one  plate 
of  the  condenser  Cs.  During  the  next  positive  loop  of  the  incoming 
wave  the  grid  attracts  more  electrons,  which  are  also  trapped  so 
that  they  cannot  escape  from  the  grid  during  the  succeeding 
negative  loop.  In  this  way  the  grid  builds  up  a  negative  potential 
and  the  high  frequency  potential  variations  on  the  grid  vary  around 


^  Incoming 
Oscillations, 


Oriel  Potential 


Plate  Current 


Audio  Current 
in  Telephone 


FIG.  200. 

the  mean  value  of  grid  potential  which  becomes  more  and  more 
negative  as  the  strength  of  the  incoming  oscillations  incr^ses. 
This  reduces  the  plate  current,  and  if  the  condenser  Cs  and  the 
insulation  of  the  part  of  the  circuit  comprising  Cs  and  the  grid 
were  perfect  the  plate  current  would  be  permanently  reduced, 
and  this  would  make  the  tube  inoperative.  To  prevent  this  a 
high  resistance  leak  Rs  is  shunted  across  the  condenser  CS)  its  value 
being  so  proportioned  that  the  electrons  cannot  leak  off  through 
this  resistance  to  any  appreciable  extent  in  a  time  comparable 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    335 

with  the  period  of  the  high  frequency  oscillations  but  do  leak  off 
in  a  time  which  is  of  the  order  of  magnitude  of  the  low  frequency 
variations  of  the  amplitude  of  the  high  frequency  oscillations. 
The  result  is  that  the  potential  of  the  grid  takes  such  values  as 
represented  by  the  second  curve  of  Fig.  200.  This  causes  the  plate 
current  wave  to  assume  the  shape  shown  in  the  diagram.  The 
high  frequency  variations  in  the  plate  circuit  pass  through  the 
condenser  Ci  (Fig.  198)  inserted  in  the  output  circuit  and  the 
current  in  the  telephone  receiver  takes  the  shape  shown  by  the 
bottom  curve  of  Fig.  200. 1 

In  order  to  secure  the  best  results  with  this  type  of  circuit 
it  is  necessary  to  operate  on  that  part  of  the  grid  voltage,  grid  cur- 
rent characteristic  which  shows  the  greatest  curvature,  and  simul- 
taneously adjust  the  plate  potential  to  such  a  value  that  the 
operating  point  on  the  plate  current,  grid  potential  characteristic 
lies  in  the  region  where  this  characteristic  is  steepest.  This 
usually  requires  that  the  grid  be  maintained  at  a  positive  potential 
with  respect  to  the  negative  end  of  the  filament.  The  simplest 
way  to  secure  this  is  to  connect  the  grid  circuit  to  the  positive  end 
of  the  filament  as  shown  in  Fig.  198  instead  of  to  the  negative  end 
as  is  commonly  done  in  other  circuits.  This  makes  the  filament 
negative  with  respect  to  the  grid,  the  average  potential  difference 
between  them  being  in  the  neighborhood  of  the  value  where  the 
grid  current  characteristic  has  its  greatest  curvature.  The  best 
value  for  the  capacity  Cs  usually  lies  between  about  150-500 
micro-microfarads  while  the  leak  resistance  Rs  should  be  of  the 
order  of  two  megohms. 

If  the  detecting  current  be  measured  as  a  function  of  the 

effective  voltage  Ey={—+Eg-\-e}  a  curve  is  obtained  like  that 

shown  in  Fig.  201.  When  the  blocking  condenser  is  not  used  we 
have  seen  the  relation  between  detecting  current  and  effective 
voltage  gives  a  maximum  as  shown  in  Fig.  197. 

104.  Method  of  Measuring  the  Detecting  Current.  The  meas- 
urement of  the  detecting  current  under  conditions  approximating 
those  met  with  in  practice  has  always  been  a  difficult  matter  because 
it  involves  the  measurement  of  very  small  alternating  currents. 
Their  values  under  practical  conditions  range  from  about  10~6 
ampere  down  to  10~8  ampere  and  sometimes  less.  This  makes  it 
1 E.  H.  ARMSTRONG,  El.  World,  Vol.  64,  p.  1149,  1914. 


336 


THERMIONIC  VACUUM  TUBE 


entirely  impossible  to  use  hot  wire  instruments.  The  telephone 
receiver  is  a  very  sensitive  device  for  indicating  small  alternating 
currents,  but  does  not  directly  give  a  measure  of  the  value  of  the 
currents  in  the  receiver.  The  audibility  method,  which  will  be 
discussed  later  on,  has  been  suggested  to  measure  detecting  cur- 
rents with  a  telephone  receiver.  It  consists  in  shunting  the  tele- 
phone receiver  with  a  variable  resistance  and  adjusting  this 
resistance  until  the  current  in  the  telephone  receiver  is  just  large 
enough  to  make  it  possible  to  discriminate  between  the  dots  and 
dashes  of  the  incoming  signals.  The  ratio  of  the  total  current 
in  the  receiver  and  shunt  resistance,  that  is,  the  detecting  current 


Plctt-e  Vol-t-age 

FIG.  201. 

to  the  current  in  the  receiver  alone,  measures  what  is  known  as  the 
"  audibility."  This  method  is  not  very  reliable,  and  its  accuracy 
depends  to  a  large  extent  on  the  conditions  under  which  the 
measurements  are  made  (see  Section  108). 

The  following  method  requires  only  that  two  notes  of  the  same 
pitch  be  adjusted  to  equal  intensities.1  It  is,  comparatively 
speaking,  very  accurate,  and  does  not  depend  nearly  so  much  on 
the  conditions  under  which  the  measurements  are  made.  The 
principle  of  this  method  can  be  explained  with  reference  to  Fig. 
202.  The  incoming  high  frequency  oscillations  are  impressed 
on  the  grid  in  the  usual  way.  In  order  to  measure  the  small 

1  H.  J.  VAN  DER  BIJL,  Phys.  Rev.,  Vol.  13,  p.  311,  1919;  Proc.  Inst.  Radio 
Engineers,  Vol.  7,  p.  603,  1919. 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    337 

detecting  current  in  the  output  of  the  detector  we  use  a  generator 
£7,  giving  a  note  of  the  same  pitch  as  that  of  the  detecting  current, 
and  then  attenuate  the  current  from  the  generator  by  means  of  a 
receiver  shunt  S  until  the  current  id  has  the  same  value  as  the 
detecting  current  delivered  by  the  tube.  W  is  a  switch  whereby 
the  telephone  receiver  can  be  connected  either  to  the  output  of 
the  tube  or  to  the  output  of  the  generator.  The  shunt  and  series 
resistances  of  the  receiver  shunt  are  adjusted  until  the  tone  heard 
in  the  receiver  is  of  the  same  intensity  for  both  positions  of  the 
switch  W.  The  receiver  shunt  has  been  described  in  Section  72. 


FIG.  202. 

The  shunt  and  series  resistances  are  varied  in  definite  steps  by  the 
simple  operation  of  turning  a  dial,  these  steps  being  so  propor- 
tioned that  the  impedance  in  the  output  of  the  generator  U 
remains  constant  for  all  adjustments  of  the  shunt.  The  current 
ii  delivered  by  the  generator  into  this  impedance  is  so  large  that 
it  can  easily  be  measured  with  a  hot  wire  instrument  A  such  as  a 
thermo-couple.  It  was  shown  in  Section  72  that  the  relation 
between  the  current  i\  and  the  branch  current  id  flowing  through 
the  receiver  is 


(28) 


where  a  is  the  constant  of  the  receiver  shunt  and  d  expresses  the 
current  attenuation  produced  by  the  shunt  in  terms  of  length  of 


338  THERMIONIC  VACUUM  TUBE 

the  cable  or  line  having  an  attenuation  constant  equal  to  a  per 
unit  length.  For  the  standard  cable  of  reference  commonly 
used  in  telephony  a  =  0.109  per  mile,  d  being  expressed  in  miles 
(see  Section  72).  Expressing  the  above  equation  in  common 
logarithms  we  get 

log^=logn--, (29) 


where 

2.303 


=  21.13.    .....     (30) 


Now  ii  is  measured  by  means  of  the  instrument  A,  and  d  is  a 
known  value  depending  on  the  adjustment  of  the  receiver  shunt 
in  the  manner  explained  in  Section  72;  hence,  if  the  shunt  be  so 
adjusted  that  the  tone  in  the  receiver  is  of  the  same  intensity  for 
both  positions  of  the  switch  W  we  can,  from  the  above  equation, 
obtain  the  detecting  current  id. 

The  impedance  of  the  telephone  receiver  should,  of  course, 
have  such  a  value  that  the  best  operation  is  obtained.  If  neces- 
sary, we  can,  to  secure  this,  insert  a  transformer  between  the  tele- 
phone receiver  and  the  output  of  the  tube.  Furthermore  the 
detecting  current  depends  on  the  value  of  the  voltage  variation 
impressed  on  the  grid  and  upon  the  extent  to  which  the  incoming 
wave  is  modulated.  The  R.M.S.  value  of  the  voltage  can  be 
measured  by  means  of  a  resistance  r  and  an  a-c.  galvanometer 
G  as  shown,  for  example,  in  Fig.  208.  When  comparing  tubes  for 
their  operation  as  detectors  the  input  need  not  be  measured,  nor 
is  it  necessary  to  insure  that  the  incoming  wave  is  completely 
modulated  as  long  as  these  quantities  remain  the  same  throughout 
the  measurements.  When  measuring  the  detecting  efficiency  of 
the  tube,  however,  it  is  necessary,  as  will  be  explained  later  on, 
to  measure  these  quantities. 

This  method  of  measuring  the  detecting  current  has  been  found 
to  be  very  useful  when  studying  the  influence  of  the  operating 
parameters  such  as  the  d-c.  plate  and  grid  potentials  on  the  detect- 
ing current.  4  When  making  such  measurements  it  is  customary  to 
express  the  detecting  current  simply  in  terms  of  the  adjustment  d 
of  the  receiver  shunt  instead  of  computing  the  actual  value  of  the 
detecting  current  from  equation  (29).  It  will  be  noticed,  how- 
ever, that  d  increases  when  the  detecting  current  decreases.  It 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    339 

is  therefore  advantageous  to  calibrate  the  receiver  shunt  in  terms 
of  D-d  when  D  is  an  arbitrary  constant. 

105.  Measurement  of  the  Detection  Coefficient.  The  method 
described  above  makes  it  possible  to  measure  the  detection  coeffi- 
cient if  the  relation  between  the  detecting  current  and  the  voltage 
impressed  on  the  input  is  known.  If  the  tube  is  used  without  a 
blocking  condenser  in  the  grid  circuit  the  detecting  current  can  be 
given  by  the  equation 

(31) 


When  the  tube  is  used  with  a  blocking  condenser  this  relation  also 
holds  fairly  accurately  provided  the  input  voltage  is  small,  gen- 
erally not  greater  than  about  half  a  volt.  If  we  put  this  value  of 
id  into  equation  (29)  we  get 

d  =  2Klog10eg+C,      ......     (32) 

where 

C  =  1C  (log  u  -log  a)  ......     (33) 

Hence  if  we  measure  the  relation  between  the  input  voltage  eg 
and  the  setting  d  of  the  receiver  shunt  we  obtain  a  straight  line 
from  the  intercept  of  which  the  detection  coefficient  a  can  be 
determined.  The  intercept  C  is  obtained  when  e0—\.  This  gives 

c 

loga  =  logz'i--^  .......     (34) 

The  detection  coefficient  can  therefore  be  obtained  to  any  desired 
degree  of  accuracy  by  taking  a  sufficiently  large  number  of  obser- 
vations of  eg  and  d. 

The  circuit  whereby  such  measurements  can  conveniently  be 
made  is  shown  in  Fig.  203.  In  this  circuit  the  source  of  audio 
frequency  current  used  to  modulate  the  high  frequency  current 
also  supplies  the  current  with  which  the  detecting  current  in  an 
output  of  the  detector  is  compared.  U  is  the  generator  of  the 
audio  frequency  currents.  This  can  be  a  vacuum  tube  oscillator 
or  a  microphone  generator  such  as  that  described  in  Section  72. 
(See  Figs.  114  and  115.)  Its  output  passes  through  a  filter  F, 
which  transmits  only  frequencies  of  about  800  cycles.  This 
current  is  sufficiently  large  to  be  measured  with  a  thermo-couple  A. 
but  after  passing  through  the  receiver  shunts  S  and  S'  it  is  atten- 
uated until  the  intensity  of  the  tone  heard  in  the  receiver  T  is  equal 


340 


THERMIONIC  VACUUM  TUBE 


to  the  detecting  current  coming  from  the  detector  tube  D.  B;- 
means  of  the  switch  W  either  the  detecting  current  or  the  current 
from  the  generator  U  can  be  passed  through  the  telephone  receiver. 
If  the  switch  is  thrown  to  the  left  the  current  from  U  passes 
directly  through  the  receiver  after  being  attenuated  by  the  receiver 
shunt.  When  W  is  thrown  to  the  right  the  output  of  U  is  im- 
pressed on  the  input  circuit  of  the  modulator  M .  This  low  fre- 


FIG.  203. 

quency  current  is  therefore  used  to  modulate  the  high  frequency 
current  also  impressed  on  the  input  of  M  and  obtained  from  the 
vacuum  tube  oscillator  0.  The  output  of  the  modulator  is 
impressed  on  the  detector  D,  the  voltage  between  filament  and 
grid  of  D  being  measured  by  means  of  the  resistance  r  and  thermo- 
galvanometer  G.  It  follows  then  from  the  equations  developed 
above  that  the  audio  frequency  output  of  the  detector  is  of  the 
same  pitch  as  the  current  supplied  by  the  generator  U,  thus 
making  the  adjustment  of  the  receiver  shunt  S  for  equal  inten- 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    341 

sities  of  these  two  notes  a  comparatively  simple  matter.  It  is 
true  that  the  output  of  the  detector  also  contains  a  note  of  double 
frequency,  as  shown,  for  example,  by  equation  (19),  but  this  double 
frequency  note  is  usually  so  weak  as  not  to  cause  any  trouble. 

The  circuit  shown  in  Fig.  203  requires  that  certain  precautions 
be  taken  to  obtain  reliable  results;  for  example,  it  is  necessary 
to  make  sure  that  the  output  impedance  of  the  generator  U 
remains  constant  for  both  positions  of  the  switch  W.  Thus,  sup- 
posing that  the  impedance  of  the  telephone  receiver  T  is  20,000 
'ohms,  it  is  necessary  to  make  the  input  impedance  of  the  trans- 
former T2  which  is  placed  in  the  input  circuit  of  the  modulator 
M  also  20,000  ohms.  This  transformer  is  usually  wound  to  have  a 
high  output  impedance  in  order  to  impress  the  highest  input 
voltage  on  the  grid  of  the  tube  to  which  it  is  connected  for  the 
lowest  amount  of  power  expended  in  the  input.  The  transformer 
TI  is  inserted  when  the  impedance  of  the  generator  U  is  different 
from  that  of  the  telephone  receiver  T.  In  order  to  adjust  the 
current  from  U  to  the  desired  value  the  primary  of  transformer  T\ 
is  shunted  with  a  resistance  and  the  connection  to  the  output  of 
the  generator  is  made  by  means  of  a  sliding  contact  as  indicated 
in  the  diagram.  The  receiver  shunt  Sf  has  a  fixed  value,  giving  an 
attenuation  equal  to  the  maximum  attenuation  given  by  the  varia- 
ble shunt  S,  and  can  be  inserted  when  the  detecting  currents  to  be 
measured  cover  a  greater  range  than  can  be  taken  care  of  by  one 
receiver  shunt.  Receiver  shunts  are  seldom  made  to  cover  a 
greater  range  of  attenuation  than  30  miles  of  standard  cable 


('4=26.3). 

\id  / 


In  making  measurements  of  this  kind  it  is  necessary  to  insure 
that  the  modulated  wave  impressed  on  the  input  of  the  detector 
is  completely  modulated.  The  necessity  for  this  can  readily  be 
seen  by  referring  to  equations  (23)  and  (24),  which  give  the  R.MJ3. 
values  of  the  detecting  current  and  the  modulated  voltage  on  the 
input  of  the  detector.  From  these  equations  it  will  be  seen  that 
for  a  constant  modulated  input  voltage  eg  the  detecting  current 
depends  on  B  and  this,  we  have  seen,  is  a  measure  of  the  extent 
to  which  the  wave  is  modulated.  This  can  also  be  seen  by  referring 
to  Figs.  188  and  191.  Two  waves  such  as  those  shown  in  these 
figures  may  have  the  same  heating  effect  as  measured,  for  example, 
by  means  of  a  resistance  and  thermo-galvanometer,  but  they  will 


342 


THERMIONIC  VACUUM  TUBE 


not  produce  the  same  detecting  effect  when  they  are  impressed  on 
the  detector.  In  the  limiting  case  in  which  the  wave  is  not  mod- 
ulated at  all  C6*=0)  the  R.M.S.  of  the  input  voltage  will  have  a 

j^ 
finite  value  — - ,  but  the  detecting  current  will  be  zero  (equation 

V2 

23).  In  order  to  insure  that  measurement  of  the  detection  co- 
efficient shall  have  any  meaning  the  extent  to  which  the  wave 
impressed  on  the  input  of  the  detector  is  modulated  must  be  kept 
constant,  and  the  simplest  way  to  do  this  is  to  completely  modulate 
the  wave,  thus  making  B=l.  This  can  readily  be  done  in  prac- 
tice in  the  following  way:  Referring  to  Fig.  204,  which  represents 


FIG.  204. 

the  relation  between  plate  current  and  grid  potential,  it  is  evident 
from  the  explanations  given  in  Section  53  that  the  intercept  OA 
which  represents  the  negative  grid  potential  necessary  to  reduce 

TjfT 

the -plate  current  to  zero  is  — .     If  we  now  apply  a  constant 

Tjl 

grid  potential  Eg=  p~  and  make  the  peak  value  of  the  low  fre- 
Zp 

quency  input  voltage  equal  to  this  quantity,  then  the  amplitude  of 
the  high  frequency  oscillations  is  reduced  to  zero  every  time  that 
the  grid  acquires  its  maximum  negative-  potential  CA  and  then 
the  wave  will  be  completely  modulated.  The  simplest  way  to 
secure  this  in  practice  is  first  to  adjust  the  negative  grid  battery 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    343 


7F       7? 

in  the  input  circuit  of  the  modulator  to  a  value  — +^;  that 

is,  to  a  value  given  by  OD  (Fig.  204),  and  then  gradually  increase 
the  strength  of  the  low  frequency  input  voltage  until  a  d-c.  meter 
placed  in  the  output  of  the  modulator  just  indicates  a  current 

flow  in  the  output  of  the  modulator.     The  peak  value  of  the 

-pi 
input  potential  is  then  equal  to  DA  or  -^-.     The  voltage  of  the  high 


10 


80 


TO 


60 


7 


Slope  =4?.? 


50 


-0.2  -0.4  -0.6  -0.8  -1.0  *ll  -1.4 

FIG.  205. 

frequency  impressed  on  the  modulator  can  also  be  measured  in 
the  same  way  and  should  in  general  be  somewhat  smaller  than  the 
low  frequency  voltage.  Finally  the  grid  battery  in  the  input 
circuit  of  the  modulator  is  adjusted  to  the  value  OC,  before  the 
measurements  on  the  detecting  current  are  undertaken. 

Fig.  205  shows  some  experimental  results  that  were  obtained 
with  the  circuit  shown  in  Fig.  203.  The  ordinates  indicate  the 
setting  of  the  receiver  shunt  for  different  values  of  the  input  volt- 
age eg,  the  logarithms  of  which  are  plotted  as  abscissae.  Accord- 


344  THERMIONIC  VACUUM  TUBE 

ing  to  equation  (32)  these  points  should  lie  on  a  straight  line. 
Furthermore,  if  equation  (31)  holds  the  slope  of  this  line  should  be 
2K;  that  Is,  42.26,  since  K  for  the  receiver  shunt  used  is  21.13. 
The  crosses  and  circles  represent  observations  made  by  two  dif- 
ferent observers  on  different  days.  The  slope  of  the  line  drawn 
through  them  is  42.2.  These  measurements  were  made  without  a 
blocking  condenser  in  the  grid  circuit  and  prove  that  in  deriving 
the  equations  in  the  previous  pages  we  were  justified  in  assuming 
that  the  power  series  given  by  equation  (2)  converges  so  rapidly 
that  we  can  neglect  quantities  of  higher  order  than  the  second, 
and  that  therefore  the  detecting  current  is  given  by  a  simple 
equation  (31). 

From  the  intercept  C  of  this  line  (log  eg  =  0),  and  the  value 
of  the  current  i\  the  detection  coefficient  can  be  obtained  directly 
with  the  help  of  equation  (34).  In  the  case  to  which  the  experi- 
mental results  given  in  Fig.  205  apply  the  current  i\  as  measured  by 
a  meter  inserted  in  the  20,000  ohm  line  was  3.10X10"3  ampere 
and  the  intercept  for  eff=l  is  40.8.  From  this  we  obtain  for  the 
detection  coefficient  a  =  36.2X10~6  amp.  (volts)2. 

106.  Detecting  Efficiency.  The  detection  coefficient  a  gives 
a  measure  of  the  audible  component  of  the  current  in  the  output 
of  the  detector  and  depends  on  the  impedance  of  the  telephone 
receiver.  It  is  therefore  not  suitable  for  expressing  the  figure 
of  merit  of  the  tube  as  a  detector.  The  impedance  of  the  tele- 
phone receiver  should  be  chosen  to  give  maximum  response.  The 
audio  frequency  output  power  is  the  quantity  which  gives  a  better 
indication  of  the  behavior  of  the  tube,  and  is  given  by  the  product 
of  the  square  of  the  detecting  current  and  the  resistance  of  the 
telephone  receiver.  The  power  developed  in  the  receiver  depends, 
of  course,  also  on  the  power  developed  in  the  input  circuit,  that 
is,  on  the  strength  of  the  incoming  oscillations,  the  figure  of  merit 
of  the  tube  as  a  detector  being  given  by  the  ratio  of  audio  frequency 
output  power  to  radio  frequency  input  power.  This  is  a  difficult 
quantity  to  measure.  It  was  shown,  for  example,  in  Section  71 
that  the  power  expended  in  the  grid  circuit  depends  on  the  con- 
stants of  the  output  circuit.  The  reaction  of  the  output  on  the 
input  circuit  through  the  electrostatic  capacities  between  the 
electrodes  of  the  tube  causes  the  tube  to  behave  as  if  it  had  an 
effective  input  impedance.  If  the  output  circuit  contains  only 
a  pure  resistance,  the  resistance  component  of  this  effective 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    345 

impedance  between  filament  and  grid  has  a  positive  value,  which, 
however,  is  usually  small  compared  with  the  reactance  component. 
If,  on  the  other  hand,  the  external  circuit  contains  an  inductive 
impedance,  the  grid  resistance  may  be  negative,  thus  giving 
rise  to  a  generation  of  power  in  the  input.  At  very  high  fre- 
quencies, it  was  shown  in  Section  71,  the  resistance  component 
of  the  input  impedance  is  practically  zero,  but  the  input  voltage 
may  be  considerably  reduced  on  account  of  the  input  circuit 
being  shunted  by  the  capacity  between  filament  and  grid.  When 
the  tube  is  used  with  a  blocking  condenser  in  the  grid  circuit  there 
is  a  convection  current  between  filament  and  grid,  thus  giving  rise 
to  an  input  resistance  which  must  be  added  to  that  caused  by  the 
inter-electrode  capacities.  If,  on  the  other  hand,  the  tube  is  used 
without  a  blocking  condenser  in  the  input  circuit,  in  which  case 
the  grid  should  be  kept  negative  with  respect  to  the  filament,  the 
input  resistance  is  due  entirely  to  the  reaction  of  the  output  cir- 
cuit to  the  input  through  the  capacities  of  the  tube,  and  can  be 
made  as  small  as  we  please  by  properly  adjusting  the  constants 
of  the  circuit.  Most  of  the  input  power  is  then  dissipated  in  the 
input  coil  and  condenser.  The  input  power  can  therefore  be 
dissipated  at  the  grid,  in  the  external  input  circuit  and  in  a  fic- 
titious input  resistance  occasioned  by  the  reaction  of  the  output 
circuit  on  the  input.  The  relative  amounts  of  power  dissipated 
in  these  parts  depends  on  the  adjustments  of  the  circuit.  It  is  for 
this  reason  usually  better  to  express  the  figure  of  merit  of  the 
device  as  a  detector  in  terms  of  the  audio  frequency  output  power 
for  a  given  high  frequency  voltage  impressed  on  the  input  because 
there  is  a  definite  relation  between  these  quantities.  The  equa- 
tions developed  in  the  previous  sections  express  the  quantities 
considered  in  terms  of  the  input  voltage,  and  therefore  hold  what- 
ever may  be  the  effect  of  the  circuit  and  the  inter-electrode 
capacities  on  the  input  power.  Expressing  the  detecting  effi- 
ciency 6  in  terms  of  the  relation  of  output  audio  frequency  power 
to  input  radio  frequency  voltage  we  have: 


(35) 


where  ro  is  the  resistance  in  the  output  of  the  detector  and  a  is  the 
detection  coefficient.  The  curve  shown  in  Fig.  205  was  obtained 
with  a  circuit  in  which  the  telephone  receiver  used  had  an  im- 


346  THERMIONIC  VACUUM  TUBE 

pedance  of  20,000  ohms  and  a  resistance  of  6400  ohms  at  about 
800  cycles  per  second.  The  detecting  efficiency  of  the  tube  on 
which  these  measurements  were  made  is  therefore  8.1X10"6 
watt  (volt)4.  The  smallest  amount  of  power  dissipated  in  this 
receiver  which  could  still  give  a  signal  that  is  barely  audible  is 
about  3  X  10~12  watt.  The  high  frequency  input  voltage  necessary 
to  give  the  least  audible  signal  with  this  particular  tube  and  tele- 
phone receiver  is  therefore  about  0.025  volt. 

These  measurements  were  made  on  a  Western  Electric  type 
VT—1  tube  (Fig.  127).  This  tube  operates  on  a  plate  voltage 
of  about  20  volts.  The  power  consumed  in  heating  its  filament 
ranges  from  2.2  to  3.5  watts.  On  account  of  the  small  amount  of 
power  involved  when  using  the  tube  as  a  detector  it  is  desirable 
to  make  the  filament  as  small  as  possible  in  order  to  reduce  the 
power  expended  in  heating  it.  The  limitation  to  the  decrease  in 
the  size  of  the  filament  is  due  mostly  to  mechanical  difficulties, 
but  smaller  types  of  tubes  have  been  developed,  of  which  the  one 
shown  in  Fig.  131  (page  244)  is  a  sample.  This  tube  only  requires 
a  small  fraction  of  a  watt  to  heat  its  filament,  the  filament  operating 
on  a  voltage  ranging  from  1.0  to  1.5  volts  so  that  it  can  be  used 
with  a  dry  cell.  The  detecting  efficiency  of  this  little  tube  was 
found  to  be  about  4.3  X 10"6  watts/ (volts)4.  A  ratio  of  two  in  the 
output  power  corresponds  to  a  difference  of  about  three  standard 
cable  miles,  which  is  not  a  big  difference.  A  difference  of  one 
standard  cable  mile  is  hardly  noticeable  unless  comparison  be 
made  directly. 

107.  Comparison  of  Detectors.  The  circuit  shown  in  Fig.  203 
is  not  always  suitable  for  use  where  a  large  number  of  tubes  are 
to  be  tested,  because  it  requires  accurate  calibration  and  careful 
adjustment  of  the  operating  parameters  such  as  the  radio  fre- 
quency voltage  impressed  on  the  input  of  the  detector,  the  audio 
frequency  current  delivered  by  the  generator  C7,  etc.  The  con- 
stancy with  which  vacuum  tubes  can  be  made,  however,  makes  it 
possible  to  test  tubes  by  comparison  methods  that  are  simple  to 
operate.  The  tubes  to  be  tested  can  then  be  compared  with  a 
standard  tube  that  was  carefully  calibrated  by  means  of  such  a 
circuit  as  shown  in  Fig.  203.  If  a  tube  is  well  evacuated  it  will 
retain  constant  operation  over  a  considerable  length  of  time. 
The  writer  has,  for  example,  used  a  "  standard  "  Western  Electric 
tube  whose  detecting  efficiencj'  did  not  change  to  any  noticeable 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    347 

extent  in  the  course  of  about  a  year,  during  which  time  it  was  in 
frequent  use. 

A  simple  circuit  whereby  detectors  can  be  compared  is  shown  in 
Fig.  206.  The  input  voltage  can  be  adjusted  to  the  desired  value 
by  adjusting  the  resistance  r  and  need  not  be  known  accurately, 
it  being  sufficient  to  know  that  it  lies  within  the  range  of  voltages 
used  in  practice.  By  means  of  the  switch  W  either  one  of  the 
detectors  can  be  inserted  in  the  circuit  and  the  receiver  shunt  S 
adjusted  until  the  note  in  the  receiver  T  has  the  same  intensity 
for  both  positions  of  the  switch  W.  The  key  K  serves  to  throw 


FIG.  206. 

the  shunt  into  or  out  of  the  circuit  according  as  W  connects  the 
tube  of  higher  or  lower  efficiency.  The  capacity  C  is  a  radio  fre- 
quency leak  and  the  output  circuit  is  connected  across  the  choke 
coil,  which  insures  that  the  d-c.  potential  on  the  plate  remains 
constant  for  all  adjustments  of  the  resistances  of  the  receiver 
shunt  S. 

If  id  and  i'a  be  the  detecting  currents  obtained  from  the  tubes 
I  and  II,  and  a  and  a',  their  detection  coefficients,  then  since  the 
input  voltage  is  the  same  for  both,  we  have 


d  —  d' 


(36) 


where  d  and  d'  are  the  adjustments  of  the  shunt  in  units  depending 
on  the  units  of  K.     The  detecting  efficiency  of  the  tube  under  test 


348  THERMIONIC  VACUUM  TUBE 

is  then  given  by 


(37) 


Fig.  207  shows  the  complete  circuit  as  it  can  be  used  for  com- 
paring detectors  in  practice.  This  circuit  contains  an  amplifier 
tube  connected  to  the  output  of  the  detectors.  When  the  detector 
is  to  be  used  in  practice  with  an  amplifier  such  a  circuit  is  desirable 
to  insure  that  the  detector  is  tested  under  conditions  approximating 
as  closely  as  possible  to  practical  conditions.  If  the  detector  is 
not  to  be  used  with  an  amplifier  the  receiver  shunt  and  telephone 


FIG.  207. 

receiver  can  be  connected  directly  to  the  output  of  the  detector 
as  shown  in  Fig.  206.  To  obtain  a  modulated  high  frequency  test 
wave  would  ordinarily  require  a  radio  frequency  oscillator,  an 
audio  frequency  oscillator  and  a  modulator,  but  in  comparing 
detectors  it  is  not  necessary  that  the  wave  be  completely  modu- 
lated, and  under  such  conditions  a  modulated  high  frequency  wave 
can  be  obtained  very  easily  by  means  of  a  vacuum  tube  oscillator 
and  a  microphone  generator  such  as  the  one  described  in  Section 
72.  In  Fig.  207  U  is  the  microphone  generator,  the  carbon  button 
of  which  is  inserted  directly  in  the  oscillation  circuit  C\L\.  When 
this  generator  is  in  operation  the  resistance  of  the  carbon  button 
varies  periodically  at  an  audio  frequency,  thus  causing  audio 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    349 

frequency  variations  in  the  amplitude  of  the  radio  frequency 
oscillations  produced  by  the  oscillator  tube.  Modulation  pro- 
duced in  this  way  is  not  complete,  but  in  this  case  complete  modu- 
lation is  not  necessary  because  the  input  is  the  same  for  both 
tubes.  The  condenser  €2  in  the  output  of  the  detector  serves  as  a 
high  frequency  leak  and  the  resistance  r  is  inserted  to  prevent  the 
grid  from  acquiring  a  negative  charge.  Its  value  is  in  the  neigh- 
borhood of  2  megohms. 

108.  Audibility  Method  of  Measuring  the  Detecting  Current. 
The  audibility  or  "  shunted  telephone  "  method  has  been  fre- 
quently applied  to  the  measurement  of  the  strength  of  received 
signals  in  long  distance  radio  communication,  and  has  also  been 
used  to  obtain  an  idea  of  the  sensitiveness  of  detectors.  In  this 
method  the  telephone  receiver  is  shunted  by  means  of  a  resist- 
ance rs  which  is  adjusted  until  the  signal  heard  in  the  receiver  is 
just  barely  audible.  If  id  is  the  detecting  current  and  io  the 
least  audible  current  in  the  receiver,  then  the  "audibility"  is 
given  by 


(38) 


where  ZQ  is  the  impedance  of  the  receiver.  In  using  a  method 
like  this  it  must,  of  course,  be  remembered  that  ZQ  cannot  be 
replaced  simply  by  the  resistance  of  the  receiver  as  is  sometimes 
done,  but  the  reactance  and  the  motional  impedance  of  the 
receiver  must  be  taken  into  consideration. 

This  method  is  open  to  other  serious  objections.  In  the  first 
place,  it  is  liable  to  considerable  error  because  the  measurement  of 
least  audible  signals  is  made  difficult  by  the  influence  of  extraneous 
noises  such  as  room  noises  and  static.  The  least  audible  current 
depends  furthermore  to  an  appreciable  extent  upon  the  condition  of 
the  observer,  so  that  the  current  necessary  to  give  the  least  audible 
signals  will  vary  from  time  to  time  even  with  the  same  observer. 
These  disadvantages  make  the  method  unreliable  for  purposes  of  de- 
termining the  detecting  efficiency  of  a  tube  with  any  degree  of  accu- 
racy. Secondly,  the  way  in  which  the  audibility  method  is  ordina- 
rily used,  does  not  make  provision  for  the  Change  in  the  effective 
impedance  of  the  output  circuit  to  the  detecting  current  when  the 
shunt  resistance  rs  is  varied.  This  would  give  misleading  results, 
since  the  detecting  current  depends  upon  the  relative  values  of  the 


350  THERMIONIC  VACUUM  TUBE 

internal  output  impedance  of  the  tube  and  the  external  impedance 
in  the  output  circuit.  It  is  therefore  necessary  in  all  measure- 
ments of  this  kind  to  adjust  these  impedances  properly  and  keep 
them  constant  throughout  the  measurements.  If  the  audibility 
method  is  to  be  used  the  "  audibility  box  "  should  be  so  designed 
that  any  variation  in  the  shunt  resistance  is  accompanied  by  an 
addition  or  subtraction  of  an  equivalent  resistance  so  as  to  keep 
the  total  impedance  of  the  circuit  constant.  This  can  be  done 
with  the  scheme  that  will  now  be  described.  This  scheme  was 
used  by  the  writer  1  to  determine  to  what  extent  the  audibility 
method  may  give  reliable  results  if  precautions  are  taken  to  elim- 
inate sources  of  error  other  than  those  which  depend  only  on  the 
psychological  and  physiological  influences  on  the  observer.  The 
fact  that  the  current  necessary  to  give  the  least  audible  signal 
has  different  values  for  different  observers  and  is  therefore  incapa- 
ble of  objective  determination  does  not  of  itself  rule  out  the 
audible  method  for  the  measurement  of  signal  strength,  since 
the  detector  set  could  first  be  calibrated  by  determining  the 
audibility  for  known  input  signals  and  then  used  by  the  same 
observer  to  make  the  final  measurements.  Hence  assuming  that 
extraneous  noises  could  effectively  be  cut  out,  the  possibility  of 
adapting  this  method  to  such  measurements  would  depend  upon 
the  extent  to  which  the  observer's  conception  of  least  audible 
signal  remains  constant  during  the  time  that  elapses  between  his 
calibration  of  the  set  and  the  making  of  the  final  measurements. 
It  is  hardly  necessary  to  say  that  the  whole  set  must  remain 
unchanged,  especially  the  tube  and  the  telephone  receiver. 

The  circuit  whereby  the  least  audible  signal  can  be  studied 
under  constant  circuit  conditions  is  shown  in  Fig.  208.  To  cut 
down  the  current  in  the  telephone  receiver  a  receiver  shunt  is  used 
such  as  that  described  in  Section  72.  This  shunt  contains  a 
series  and  shunt  resistance,  both  of  which  are  variable,  instead 
of  simply  a  shunt  resistance.  The  receiver  shunt  is  so  calibrated 
that  the  series  and  shunt  resistances  are  changed  simultaneously 
in  such  a  way  that  the  total  impedance  to  the  detecting  current 
in  the  output  circuit  of  the  tube  remains  constant.  A  choke  coil 
L  by-passes  the  direct  current  in  the  plate  circuit  and  insures  that 
the  d-c.  potential  of  the  plate  remains  constant  for  all  adjustments 
of  the  receiver  shunt.  This  is  necessary  because  the  shunt  is  so 
1  H.  J.  VAN  DER  BIJL,  Proc.  I.R.E.,  Vol.  7,  p.  624, 1919. 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    351 

designed  that  for  all  its  adjustments  the  impedance  of  the  output 
circuit  remains  constant,  but  the  resistance  does  not  remain  con- 
stant, and  therefore  if  the  choke  coil  and  capacity  were  omitted, 
thus  making  it  necessary  for  the  d-c.  plate  current  to  pass  through 
the  shunt,  the  potential  of  the  plate  would  be  different  for  dif- 
ferent adjustments  of  the  receiver  shunt.  It  is  very  important 
to  keep  the  relation  between  the  impedances  constant  because  the 
detecting  current  depends  very  markedly  on  the  value  of  the 
external  impedance  in  the  plate  circuit. 

The  wave  impressed  on  the  input  of  the  detector  can  be  a  spark 
signal  wave  or  a  modulated  wave  which  is  interrupted.  For  test 
purposes  such  a  wave  can  easily  be  obtained  with  an  arrangement 
such  as  that  shown  in  Fig.  207,  where  the  oscillator  tube  0  and  the 
microphone  generator  U  together  form  a  simple  system  for  pro- 


FIG.  208. 

ducing  modulated  waves.  The  output  of  this  oscillator  system 
can  then  be  passed  through  an  omnigraph  to  produce  the  signals. 
The  R.M.S.  of  the  input  can,  as  before,  be  measured  by  means  of  a 
resistance  r  and  galvanometer  G  (Fig.  208). 

The  use  of  the  receiver  shunt  makes  it  possible  to  express  the 
audibility  in  a  simple  way.     If  u  be  the  detecting  current  and  io 

the  least  audible  current  in  the  receiver  the  audibility  4-  can  be 

*o 

expressed  in  miles  of  standard  cable  by  making  use  of  the  equa- 
tions developed  in  Sections  104  and  105.  Thus,  taking  the  case 
in  which  the  detecting  current  is  proportional  to  the  square  of  the 
input  voltage,  we  obtain 

d  =  2Klogeg+K(\oga-\ogio),   ....     (39) 

which  thus  gives  a  linear  relation  between  the  logarithm  of  the 
input  voltage  and  the  audibility  when  expressed  in  terms  of  miles 


352 


THERMIONIC  VACUUM  TUBE 


d  instead  of  current  ratio,  the  relation  between  d  and  the  cur- 
rent ratio  being  given  by  equation  (28).  The  intercept  of  this 
line,  obtained  by  putting  eg  =  0,  is 


K  (log  a -log  to), 


(40) 


and  gives  a  measure  of  the  audibility  efficiency  of  the  tube  ex- 
pressed in  miles/ (volt)2. 

The  simple  linear  relation  (39)  makes  it  possible  to  obtain  the 
audible  efficiency  as  an  average  of  a  large  number  of  observa- 


19 

/ 

Audibility  in  Miles  of  Standard  Cable 

IO  o  e>  o  o  S  S 

0 

/ 

S/ 

/° 

°y 

'0  Slop 

e=4l 

</ 

/ 

/ 

-S 

/ 

.8          -1.6           -1.4           -I.Z           -1.0          -0.8           -0.6           -0.4           -0.2            0 
Log  Clnput  Voltage) 

FIG.  209. 

tions.  A  number  of  such  observations  plotted  against  the  logar- 
ithm of  the  corresponding  input  voltages  are  shown  in  Fig.  209. 
The  individual  points  vary  considerably  compared  with  the  obser- 
vations shown  in  Fig.  205,  which  were  obtained  with  'the  other 
method  discussed  above,  but  these  points  are  evenly  grouped 
about  a  straight  line  the  slope  of  which  is  about  41.  The  theo- 
retical value  of  the  slope  is  42.26.  The  slope  of  the  line  depends 
upon  the  attenuation  constant  of  the  shunt  and  the  simple  quad- 
ratic relation  (17)  between  the  detecting  current  and  the  input 
voltage.  The  intercept  of  this  line,  on  the  other  hand,  is  influ- 


'DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    353 

enced  also  by  the  detection  coefficient  a  of  the  tube  and  by  what 
constitutes  the  least  audible  signal  IQ  for  the  particular  telephone 
receiver  used  and  for  the  observer  at  the  particular  time  of  making 
the  measurements.  The  attenuation  factor  K  of  the  receiver 
shunt  is  determined  merely  by  resistances  of  wires,  and  does,  of 
course,  not  vary  to  any  noticeable  extent.  By  using  a  good  tube 
for  which  the  detection  coefficient  remains  constant  such  measure- 
ments can  therefore  be  used  to  give  an  idea  of  the  extent  to  which 
the  determination  of  least  audible  signal  depends  on  the  observer 
and  how  the  "  least  audible  current "  varies  from  one  observer  to 
another.  Experiments  conducted  by  the  writer  1  along  this  line 
and  in  which  observations  were  made  by  four  different  observers 
over  a  period  of  eight  days,  the  total  number  of  observations 
being  something  like  350,  showed  that  the  maximum  variation 
in  the  audibility  expressed  as  a  current  ratio  for  the  four  observers 
was  almost  600  per  cent,  but  the  variation  in  the  measurements  of 
each  observer  over  a  period  of  about  a  week  averaged  about  100 
per  cent.  It  must,  however,  be  remembered  that  a  variation  of 
say  50  per  cent  in  the  audibility  expressed  as  a  current  ratio  is  not 
a  serious  matter.  In  fact,  such  a  variation  would  hardly  be 
noticeable  unless,  of  course,  the  comparison  be  made  directly. 
The  more  satisfactory  way  of  expressing  the  audibility  is  to  use 
the  logarithmic  scale,  that  is,  to  express  it  in  terms  of  length  d  of 
cable  or  line.  When  the  audibility  is  expressed  in  this  way  the 
maximum  variation  observed  was  28  per  cent. 

These  measurements  have  also  shown  that  for  a  tube  like 
the  average  VT—  1  the  smallest  input  voltage  (R.M.S.)  that  can 
just  give  the  least  audible  signal  is  of  the  order  of  0.03  volt.  If 
the  incoming  signals  are  weaker  an  amplifier  must  be  attached 
to  the  output  of  the  detector.  The  minimum  input  voltage 
depends,  of  course,  also  on  the  sensitiveness  of  the  telephone 
receiver  used  in  making  the  measurements. 

109.  Heterodyne  Reception  with  the  Audion.  The  heterodyne 
method  of  reception  consists  in  supplementing  the  incoming  high 
frequency  currents  with  a  locally  generated  current  of  a  frequency 
which  differs  from  that  of  the  received  current  by  an  amount  which 
lies  within  the  audible  range.  This  method  makes  it  possible  to 
detect  continuous  waves  of  constant  amplitude,  and  furthermore 
greatly  increases  the  strength  of  the  audible  current  in  the  receiver 

1  Loc.  cit.,  p.  623. 


354  THERMIONIC  VACUUM  TUBE 

placed  in  the  output  of  the  detector.  The  manner  in  which  the 
heterodyne  method  increases  the  detecting  current  can  readily 
be  seen.  Thus  taking  the  case  in  which  the  detecting  current  is 
proportional  to  the  square  of  the  input  voltage  as  given  by  equa- 
tion (17),  if  we  impress  on  the  input  circuit  of  the  detector  two 
high  frequency  voltages  e\  sin  pt  and  62  sin  qt,  the  detecting  cur- 
rent is  given  by 

/d  =  a(ei  sin  pt+e2  sin  qt)2  .....     (41) 


On  evaluating  this  expression  and  dropping  all  terms  representing 
frequencies  that  lie  outside  of  the  audible  range  we  get 

Id  —  aeie2  cos  (p—q)t  ........     (42) 

If  £-  is  100,000  cycles,  for  example,  and  ^  99,000  cycles,  then  the 
ZTT  2w 

above  expression  represents  a  current  having  a  frequency  of  a 
thousand  cycles  per  second  and  is  therefore  audible.  Further- 
more, the  locally  generated  voltage  62  can  be  made  much  larger 
than  the  voltage  e\  of  the  incoming  waves,  and  therefore  the 
detecting  current  can  be  very  much  increased. 

Equation  (42)  holds,  strictly  speaking,  only  when  the  grid  is 
maintained  at  a  sufficiently  high  negative  potential  with  respect 
to  the  filament  to  prevent  any  electrons  from  flowing  from  the 
filament  to  the  grid.  Under  these  conditions  the  detecting  cur- 
rent has  been  found  to  be  proportional  to  the  product  61^2  of  the 
input  voltages.  These  measurements  can  be  made  with  a  circuit 
like  that  shown  in  Fig.  202.  Since  the  detecting  current  in  this 
case  can  easily  be  made  so  large  that  it  can  be  measured  with  a 
thermo-couple  without  increasing  the  strength  of  incoming  signal 
beyond  a  practical  range,  it  is  also  possible  to  use  a  circuit  like 
that  shown  in  Fig.  210.  A  high  frequency  voltage  of  small  ampli- 
tude is  impressed  at  A  and  represents  the  incoming  signal  wave 
received  by  the  antenna.  A  locally  generated  high  frequency  is 
impressed  at  B,  and  is  adjustable.  By  means  of  the  switch  W  a 
thermo-galvanometer  G  can  be  used  to  measure  the  detecting  cur- 
rent id  and  the  effective  voltage  impressed  on  the  grid.  In  using 
such  a  method  it  is,  however,  necessary  to  insure  that  when  the 
switch  connects  the  galvanometer  to  the  output  of  the  tube  the 
impedance  of  the  input  circuit  is  not  changed.  This  can  be 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    355 

taken  care  of  by  inserting  a  resistance  r\~\-r2  on  the  input  side 
having  a  value  equal  to  the  resistance  of  the  galvanometer  plus 
the  resistance  r2,  which  may  be  added  in  the  galvanometer  circuit 
to  measure  the  input'  voltage.  The  key  K  should  therefore  be 
opened  when  the  galvanometer  is  connected  to  the  input  and  closed 
when  it  is  connected  to  the  output  of  the  tube. 

Results  obtained  by  my  associate,  Mr.  R.  H.  Wilson,  are  given 
in  Fig.  211,  which  shows  the  relation  between  the  detecting 
current  measured  in  this  way  and  the  product  #162  of  the  input 
voltages  for  three  different  values  of  the  voltage  of  the  battery 
in  the  grid  circuit.  In  general,  the  curves  do  not  coincide  on  the 


FIG.  210. 

straight  part.  The  values  shown  in  this  figure  were  obtained  by 
reducing  the  observations  to  a  common  value  to  superimpose  the 
curves  on  one  another.  For  each  of  these  grid  voltages  the  voltage 
of  the  plate  battery  was  adjusted  to  maintain  the  space  current 
constant.  In  other  words,  the  operating  points  on  the  charac- 
teristics for  the  three  cases  are  shown  at  A,  B  and  C,  of  Fig  212, 
where  the  three  curves  represent  the  characteristics  obtained 
with  three  different  plate  voltages.  Fig.  211  shows  a  linear 
relation  between  the  detecting  current  and  the  product  £162  up 
to  a  value  of  this  product  depending  on  the  value  of  the  d-c. 
grid  potential  chosen.  The  point  at  which  the  observed  detecting 
current  deviates  from  the  straight  line  is  obtained  when  the  sum 


356 


THERMIONIC  VACUUM  TUBE 


of  the  peak  values  of  the  voltages  impressed  on  the  grid  becomes 
greater  than  the  negative  d-c.  grid  potential  so  that  current 
begins  to  flow  in  the  grid  circuit,  thus  causing  a  waste  of  power. 
It  is  seen  therefore  that  the  strength  of  the  signal  can  be  increased 
by  increasing  the  negative  grid  potential  and  at  the  same  time 


00x10 


1400 


1200 
|~IOOO 

4-T 

i 


! 

too 


400 


eoo 


EC--6.I8 
J^~  Volts 


Vo/fe 


Volts. 


0.6  0.8 

e,e2 

FIG.  211. 


i.o 


14 


increasing  the  plate  potential.  In  the  case  of  the  lower  char- 
acteristic the  input  voltage  cannot  much  exceed  the  value  OA, 
but  in  the  case  of  the  characteristic  with  the  high  plate  potential 
the  input  can  be  made  as  large  as  OC,  without  allowing  electrons 
to  flow  from  filament  to  grid.  Increasing  the  sum  of  the  input 
voltages  superimposed  on  the  negative  grid  potential  beyond 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    357 


the  value  necessary  to  reduce  the  current  to  zero  does  not  seem  to 
cause  a  deviation  from  the  linear  relation  between  the  detecting 
current  and  the  product  of  the  input  voltages.  The  observations 
shown  in  Fig.  211  represent  in  a  general  way  what  is  to  be  expected, 
although  it  is  possible  that  with  different  tubes  and  different 
circuits  peculiarities  of  behavior  causing  a  deviation  from  such 
a  simple  relation  can  manifest  themselves. 

When  the  heterodyne  method  of  reception  is  used  with  a 
blocking  condenser  in  the  grid  circuit  the  relation  between  the 
detecting  current  and  the  input  voltage  is  not  as  simple  as  when 
the  blocking  condenser  is  omitted.  In  general,  the  detecting 


FIG.  212. 

current  has  a  maximum  value  at  a  value  of  the  product 
depending  on  the  operating  parameters  of  the  tube,  and  the 
maximum  detecting  current  is  generally  larger  than  when  the 
blocking  condenser  is  omitted.  Fig.  213  shows  the  general 
relation  between  the  detecting  current  and  the  product  of  the  input 
voltages.  The  three  curves  are  obtained  with  three  different 
values  of  the  d-c.  grid  potential,  the  plate  battery  voltage  having  a 
constant  value  in  this  case  of  22  volts.  These  measurements 
were  made  with  a  standard  VT—l  tube.  It  will  be  noticed 
that  the  best  reults  are  obtained  when  the  grid  is  kept  positive 
with  respect  to  the  filament.  This  is  in  accordance  with  the  expla- 
nation given  in  Section  103,  where  it  was  shown  that  detection 
with  a  blocking  condenser  in  the  grid  circuit  depends  on  the 


358 


THERMIONIC  VACUUM  TUBE 


curvature  of  the  grid  current  characteristic.  The  detecting 
current  increases  with  increase  in  the  curvature  of  this  char- 
acteristic and  also  becomes  greater  the  steeper  the  plate  cur- 
rent grid  potential  characteristic  is.  If  the  relation  between 
the  maximum  detecting  current  for  a  constant  voltage  of  the  grid 
battery  be  plotted  as  a  function  of  the  plate  battery  voltage  a 
curve  is  obtained  such  as  that  obtained  in  Fig.  214.  This  curve 
shows  a  rapid  increase  in  detecting  current  with  increase  in  the 
plate  voltage  until  a  plate  battery  voltage  of  about  34  volts  is 
reached.  The  bend  in  the  curve  at  higher  voltages  is  due  to  the 


2.6 


FIG.  213. 

plate  current  characteristic  becoming  less  steep  when  voltage 
saturation  is  approached. 

110.  Zero  Beat  or  Homodyne  Method  of  Receiving  Modulated 
Waves.  The  heterodyne  method  of  reception  can  be  used  when 
receiving  continuous  unmodulated  waves,  as  is  done  in  radio 
telegraphy,  but  cannot  be  applied  to  the  reception  of  modulated 
waves  as  in  radio  telephony.  In  this  case  we  can  make  use  of  the 
simple  systems  which  consist  in  first  detecting  the  modulated 
high  frequency  waves  and  then  amplifying  the  low  frequency 
component  with  the  amplifier  tube  attached  to  the  output  of  the 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    359 


detector.  Or  the  received  high  frequency  currents  can  first  be 
amplified  and  then  detected.  This  latter  scheme  has  the  advan- 
tage of  greater  sensitivity  because  the  voltage  impressed  on  the 
input  of  the  detector  which  is  connected  to  the  output  of  the 
amplifier  is  amplified  and  the  detecting  current  we  have  seen  above 


-6 
100x10 


90 


80 


I 


60 


50 


40 


Grid  Potential =3.0 Vo/rs 


16 


Z4  32 

Plcxte  VolU 

FIG.  214. 


40 


48 


56i 


is  practically  proportional  to  the  square  of  the  input  voltage. 
Such  a  circuit,  however,  has  a  tendency  to  sing.  This  can  usually 
be  prevented  by  feeding  back  part  of  the  output  energy  to  the 
input  circuit  so  that  the  currents  in  the  feed-back  and  input  coils 
cause  potential  variations  on  the  grid  that  are  180°  out  of  phase. 


360  THERMIONIC  VACUUM  TUBE 

There  is,  however,  another  method,  whereby  modulated  high 
frequency  currents  can  be  received.  This  method  is  analogous 
to  the  heterodyne  method  except  that  the  local  source  of  high 
frequency  currents  has  a  frequency  equal  to  that  of  the  received 
carrier  wave  instead  of  differing  from  it  by  an  amount  which  lies 
within  the  audible  range.  Thus,  if  we  impress  on  the  input  of 
the  detector  a  modulated  high  frequency  voltage  A  sin  pt 
(1-f  J5  sin  qt)  and  also  a  continuous  high  frequency  voltage 

T) 

e  sin  pt  having  the  same  frequency  —  as  the  carrier  component 

Zir 

of  the  received  wave,  then  it  follows,  similarly  to  the  ca.ses  con- 
sidered above,  that  the  output  of  the  detector  contains  a  current 
of  the  same  frequency  as  the  current  with  which  the  carrier  wave 
was  modulated  at  the  transmitting  station.  This  means  of  receiv- 
ing has  been  termed  the  "  homodyne  method."  It  is  hardly 
necessary  to  say  that  when  using  this  method  of  reception  care 
must  be  taken  to  keep  the  two  high  frequencies  very  closely  in 

tune.    For  example,  ^-  in  the  above  expression  represents  speech 

2ir 

frequencies  which  only  cover  a  range  of  about  2000  cycles  per 
second.  Now,  if  the  carrier  wave  has  a  frequency  of  say  200,000 
cycles  per  second,  it  follows  that  the  locally  generated  source  of 
high  frequency  must  be  kept  at  200,000  cycles  within  a  very  small 
fraction  of  1  per  cent.  If,  for  example,  it  differs  from  this  fre- 
quency by  J  per  cent  the  locally  generated  and  the  received  cur- 
rents form  a  beat  note  of  about  500  cycles,  resulting  in  speech  which 
would  be  unintelligible.  The  way  to  tune  in  the  locally  generated 
high  frequency  is  to  listen  to  the  beat  note,  the  pitch  of  which 
changes  rapidly  as  the  two  frequencies  approach  each  other,  until 
when  they  are  exactly  alike  the  beat  note  ceases  and  intelligible 
speech  is  heard  in  the  telephone  receiver. 

111.  The  Feed-back  Receiving  Circuit.  It  will  now  be  appar- 
ent from  considerations  given  in  this  and  the  last  previous  chapter, 
that  a  very  simple  and  effective  means  of  receiving  can  be  obtained 
by  using  the  heterodyne  principle  together  with  the  amplifying 
property  of  the  tube.  In  other  words,  the  detector  tube  is  used 
to  generate  its  own  high  frequency  by  feeding  back  part  of  the 
output  energy  into  the  input.  The  circuit  arrangement  is  shown 
in  Fig.  215.  The  coil  LI,  which  forms  part  of  the  output  circuit, 
is  coupled  to  the  input  so  that  part  of  the  output  energy  is  returned 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    361 


to  the  input.  The  condenser  €2  serves  as  a  by-pass  for  the  high 
frequency  currents.  This  set  is  operated  very  simply  by  tuning 
the  input  and  varying  the  coupling  between  LI  and  the  input 
until  a  note  of  the  desired  pitch  and  intensity  is  heard  in  the 
receiver.  Measurements  that  have  been  made  on  this  kind  of 
circuit  show  that  when  the  input  voltage  impressed  on  the  grid 
reaches  a  certain  small  value  the  telephone  receiver  in  the  output 
suddenly  gives  a  relatively  loud  note,  and  generally  a  coupling 
which  is  slighly  greater  than  that  necessary  to  do  this  is  the  best. 
This  circuit  is  sometimes  referred  to  as  the  regenerative  circuit. 
It  is  also  possible  to  make  the  output  react  on  the  input  without 
direct  electromagnetic  coupling,  in  which  case  the  coupling  is 


FIG.  215. 

supplied  by  the  electrostatic  capacities  between  the  electrodes  of 
the  tube.  The  regenerative  action  of  the  tube  has  been  discussed 
in  Section  88. 

112.  Radio  Transmitting  and  Receiving  Systems.  A  large 
number  of  radio  systems  are  now  in  use  and  no  attempt  will  be 
made  to  describe  them  here.  They  can  be  understood  or  even 
designed  with  ease  once  the  principles  of  operation  of  the  tube 
are  understood.  A  circuit  which  embodies  enough  to  indicate  how 
the  tube  can  be  used  in  a  transmitting  and  receiving  system  is 
shown  in  Fig.  216.  The  circuit  to  the  left  of  the  antenna  rep- 
resents the  transmitting  circuit  and  is  practically  identical  with 
that  shown  in  Fig.  179  (page  307).  This  set  was  designed  by  the 
engineers  of  the  Western  Electric  Company  for  use  on  aeroplanes. 
The  plate  and  filament  voltages  are  supplied  b»y  a  wind-driven 


362 


THERMIONIC  VACUUM  TUBE 


generator,  the  voltage  of  which  is  regulated  by  means  of  a  ther- 
mionic valve  in  the  manner  described  in  Section  51.  When 
transmitting,  the  switch  T  is  thrown  to  the  left.  The  several 
parts  of  this  switch  are  shown  in  separate  parts  of  the  diagram 
for  the  sake  of  simplicity,  but  they  are  all  operated  from  one  lever. 
When  receiving,  the  switch  is  thrown  to  the  right.  This  cuts  out 
the  voltage  of  the  generator,  the  plates  of  the  detector  and  ampli- 
fier tubes  being  supplied  by  storage  batteries.  The  receiving  cir- 
cuit to  the  right  of  the  antenna  is  a  simple  circuit  with  inductive 
connection  between  the  tubes. 


Modulator  Test  Switch 

Input  \TRAHSMITTER    | 

Transformer    xs  .     ^ 

\  Modulator*"    *"]!*&  \ 


Amplifier 


Wind 

Driven 

Generator 

Differential 
Field 


'Moim  Field 


tency  Choke  "  Plate  Battery 

.-I---  HOTE'.  Switch  Arms  T 

Low--  Operate  from  One  Lever 

%%ry    i— -i*-* 


FlG.  216. 


A  simplified  drawing  of  such  a  receiving  system  is  shown  in 
Fig.  217.  Since  this  circuit  has  a  blocking  condenser  Cs  in  the 
input,  the  grid  is  connected  through  the  input  circuit  to  the  pos- 
itive pole  of  the  filament  battery.  €2  is  a  high  frequency  leak, 
and  the  high  resistance  leak  r  and  condenser  Cs  serve  to  main- 
tarn  the  grid  of  the  amplifier  tube  at  a  d-c.  potential  equal  to  that 
of  the  negative  end  of  the  filament.  If  necessary,  a  grid  battery 
can  be  inserted  to  keep  the  grid  of  the  amplifier  negative.  This 
circuit  shows  two  batteries,  E*,  for  supplying  the  plate  currents. 
The  second  batteiy  is  inserted  when  it  is  necessary  to  operate 
the  amplifier  at  a  higher  plate  voltage  than  the  detector. 

The  transmitting  system  shown  in  Fig.  216  has  the  advantage 
of  high  efficiency,  but,  on  the  other  hand,  the  oscillation  coil  of 
the  oscillator  forms  part  of  the  antenna,  and  therefore  the  fre- 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    363 


quency  of  the  wave  radiated  changes  with  slight  changes  in  the 
inductance  of  this  coil. 

A  system  which  is  not  subject  to  this  disadvantage  is  shown 
schematically  in  Fig.  218.  The  principle  of  this  system  consists 
in  modulating  a  low-power  radio  frequency  current  and  then 
amplifying  this  modulated  current  before  it  enters  the  antenna. 


Defector 


Amplifier   . 


FIG.  217. 

The  oscillator  0  may  be  a  vacuum  tube  oscillator  giving  the  radio 
current  at  the  desired  frequency.  The  voltage  obtained  from  this 
generator  and  the  speech  voltage  obtained  from  the  transmitter 
are  both  impressed  on  the  modulator  M.  This  part  of  the  system 
does  not  have  to  handle  large  amounts  of  power.  The  oscillator 
tube  can  be  a  small  type  of  tube  operating  on  a  plate  voltage  of 


To  Antenna 


FIG.  218. 

100  to  200  volts  or  even  less,  and  the  modulator  can  be  a  tube  of 
about  the  same  capacity.  The  speech  voltage  obtained  from  the 
telephone  transmitter  can  then  be  impressed  directly  on  the  input 
of  this  modulator  without  amplification.  The  modulated  output 
is  amplified  by  an  amplifier  A\t  and  can,  if  necessary,  be  further 
amplified  by  means  of  a  power  tube,  or  a  set  of  power  tubes  oper- 


364  THERMIONIC  VACUUM  TUBE 

ated  in  parallel.  The  output  from  these  power  tubes  is  impressed 
on  the  antenna.  It  will  be  evident  that  in  such  a  system  detuning 
of  the  antenna  does  not  change  the  frequency  of  the  radiated  wave, 
but  merely  changes  the  energy  radiated. 

This  system  is  of  great  advantage  when  using  very  short  waves 
such  as  could  be  used  in  small  portable  transmitting  sets,  in  which 
case  the  antenna  is  small  enough  to  be  carried  in  the  hand.  For 
such  short  waves  (about  20  to  100  meters)  a  change  in  the  antenna 
capacity  caused  by  the  approach  to  objects  does  not  appreciably 
change  the  frequency  of  the  radiated  wave  but  only  the  amount 
of  energy  radiated,  because  the  antenna  does  not  form  part  of 
the  oscillation  circuit  that  determines  the  frequency  of  the  oscil- 
lations. This  frequency  is  determined  by  the  constants  of  the 
oscillator  0. 

The  arrangement  shown  in  Fig.  218  embodies  the  system  that 
was  used  in  1915  by  the  American  Telephone  &  Telegraph  Com- 
pany and  Western  Electric  Company  in  transmitting  speech  from 
Arlington,  Va.,  U.  S.  A.,  to  Paris  and  Honolulu.  The  type  of 
power  tube  used  in  these  experiments  is  shown  in  Fig.  133,  page  245. 
In  these  experiments  the  waves  were  received  by  the  zero  beat 
method  and  also  by  first  amplifying  the  incoming  waves  and  then 
detecting  them. 

113.  Multiplex  Telegraphy  and  Telephony.  It  was  pointed 
out  in  Section  99  that  for  a  single  telephone  transmission  a  band 
of  frequencies  is  required  which  covers  a  range  of  about  2000 
cycles  per  second,  and  the  antenna  must  be  so  tuned  that  it  will 
transmit  about  equally  well  all  frequencies  lying  within  this 
range.  Antennae  as  commonly  used  will  usually  transmit  a  wider 
range  of  frequencies  than  that  required  for  one  telephone  trans- 
mission. This  is  especially  the  case  when  the  antenna  is  tuned 
to  a  high  frequency,  say  a  million  per  second.  The  width  of  the 
transmission  band  of  an  antenna  depends,  of  course,  on  its 
selectivity  or  sharpness  of  tuning.  If  the  transmission  band  is 
wider  than  the  range  of  frequencies  needed  for  a  telephone  trans- 
mission, it  is  theoretically  possible  to  transmit  more  than  one 
telephone  message  simultaneously  from  one  tower  by  connecting 
to  the  sending  antenna  a  number  of  transmitting  sets,  the  fre- 
quencies of  which  are  so  distributed  in  the  transmission  range 
of  the  antenna  that  the  frequency  bands  of  their  modulated  waves 
do  not  overlap,  and  to  the  receiving  antenna  corresponding 


DETECTION  OF  CURRENTS  WITH  THE  VACUUM  TUBE    365* 

receiving  sets  each  are  tuned  selectively  to  receive  the  waves  from 
one  transmitting  set.  These  frequency  bands  and  their  separation 
are  so  small  compared  with  the  frequencies  themselves,  that  it  is 
very  difficult  to  separate  them  at  the  receiving  station.  Besides, 
small  change  in  the  frequencies  of  one  or  more  of  the  trans- 
mitting sets,  coupled  to  the  same  antenna,  are  likely  to  cause 
overlapping  of  the  bands. 

Instead  of  impressing  the  modulated  waves  from  different 
transmitting  sets  of  the  same  antenna,  it  is  possible  to  modulate 
one  high  frequency  wave  with  two  or  more  different  frequency 
bands.  If  these  frequency  bands  lie  in  the  audible  range,  such  a 
system  obviously  does  not  allow  of  more  than  one  telephone 
transmission.  In  the  case  of  telegraph  transmission,  on  the 
other  hand,  the  required  frequency  band  is  much  narrower,  since 
here  we  transmit  only  one  note,  so  that  more  than  one  such  band 
lying  within  the  audible  range  could  be  used  as  messages  to  modu- 
late one  high  frequency  carrier  wave.  Such  a  modulated  wave 
can  be  passed  through  a  detector  at  the  receiving  station  and  the 
original  message  frequencies  in  the  output  of  the  detector  can  be 
separated  from  one  another  by  the  use  of  appropriate  filters  or 
selective  circuits;  but  the  output  circuit  of  the  detector  will 
contain  not  only  the  message  frequencies  with  which  the  carrier 
wave  is  modulated,  but  also  currents  of  double  the  message 
frequencies  and  currents,  having  frequencies  equal  to  their  sums 
and  differences,  and  care  must  be  taken  to  separate  the  original 
message  frequencies  from  these. 

Multiplex  transmission  has  been  realized  in  practice  to  be  a 
system  involving  what  may  be  termed  "double  modulation"  and 
"double  detection."  Here  each  of  the  various  audio  frequency 
currents  (telephone  or  telegraph  messages)  is  used  to  modulate 
one  of  a  number  of  intertnediate  or  auxiliary  frequencies  lying 
above  the  audible  range,  and  the  resulting  modulated  waves 
are  all  made  to  modulate  a  single  high  frequency  or  main  carrier 
wave. 

At  the  receiving  end  this  doubly  modulated  main  carrier 
wave  is  impressed  on  a  detector,  the  output  of  which  will  contain 
the  modulated  auxiliary  carriers.  It  will,  however,  also  contain 
a  number  of  other  frequencies,  containing  among  others  the  sums 
and  differences  of  the  modulated  auxiliary  carriers.  These  can 
be  separated  from  the  auxiliaries  themselves  by  selective  circuits 


366  THERMIONIC  VACUUM  TUBE 

if  the  frequency  bands  of  the  latter  are  not  too  close  together. 
The  modulated  auxiliary  carriers  once  separated  can  be  passed 
through  individual  detectors,  the  output  of  which  will  contain 
the  original  message  frequencies. 

Numerous  modifications  of  this  general  idea  involving  other 
special  methods  of  modulation  and  detection  are  of  course 
possible. 


CHAPTER  X 
MISCELLANEOUS  APPLICATIONS  OF  THERMIONIC  TUBES 

Besides  the  functions  that  have  been  discussed  in  previous 
chapters,  there  are  a  number  of  miscellaneous  uses  to  which  the 
thermionic  vacuum  tube  can  be  applied.  Some  of  these  will  now 
be  described.  The  list  is  not  intended  to  be  complete,  but  merely 
to  give  an  indication  of  what  can  be  done  with  a  three-electrode 
device.  Many  uses  of  the  tube  will  suggest  themselves  to  those 
who  are  acquainted  with  the  principles  of  its  operation. 

114.  The  Vacuum  Tube  as  an  Electrostatic  Voltmeter.  Three 
of  the  properties  of  the  tube,  namely,  the  unilateral  conductivity, 
amplification  and  small  power  consumption  in  the  input  circuit, 
make  it  possible  to  use  this  tube  to  measure  small  a-c.  voltages 
by  electrostatic  means.  Suppose,  for  example,  that  a  source  of 
small  a-c.  voltage  is  included  in  the  grid  circuit,  which  also  con- 
tains an  adjustable  grid  battery.  The  output  circuit  contains 
the  plate  battery  and  a  d-c.  measuring  instrument.  '  The  plate  and 
grid  batteries  are  adjusted  so  that  the  operating  point  lies  on  the 
extreme  lower  end  of  the  characteristic.  The  current  in  the  plate 
circuit  will  then  be  just  reduced  to  zero.  If  an  alternating  poten- 
tial now  be  applied  to  the  grid,  the  potential  of  the  grid  during  the 
negative  half  cycle  becomes  more  negative  than  that  maintained 
by  the  grid  battery,  and  the  current  in  the  plate  circuit  is  still  zero. 
During  the  positive  half  cycle,  however,  the  grid  becomes  less 
negative  and  a  current  impulse  is  sent  through  the  plate  circuit. 
The  result  is  a  direct  current  indicated  by  the  measuring  instru- 
ment connected  in  the  plate  circuit  of  the  tube.  The  amount  by 
which  the  negative  potential  of  the  grid  must  be  increased  to 
reduce  the  current  again  to  zero,  measures  the. peak  value  of  the 
alternating  potential  on  the  grid.1 

In  order  to  get  a  more  sensitive  device,  if  necessary,  an  ampli- 
fier could  be  added  to  the  output  of  the  tube.     A  circuit  arrange- 
1 R.  A.  HEISING,  U.  S.  patent  1232919. 
367 


368 


THERMIONIC  VACUUM  TUBE 


ment  that  can  be  used,  and  which  was  suggested  by  R.  H.  Wilson, 
is  shown  in  Fig.  219.  The  alternating  voltage  is  impressed  at  B. 
The  d-c.  grid  potential  is  supplied  by  means  of  the  batteries  Ec, 
and  the  potentiometer  arrangement  is  indicated  in  the  circuit. 
This  arrangement  makes  it  possible  to  adjust  the  grid  potential  to 
a  value  where  the  plate  current  of  the  first  tube  is  just  reduced  to 
zero.  The  output  of  the  voltmeter  tube  contains  a  high  resistance 
of,  say,  400,000  ohms,  the  grid  and  filament  of  the  amplifier  tube 
being  connected  to  the  ends  of  this  resistance.  The  amplifier  tube 
operates  on  the  steepest  part  of  its  characteristic. 

The  way  to  operate  a  vacuum  tube  voltmeter  circuit  is  to  slide 
the  contact  on  the  potentiometer  until  the  current  of  the  first  tube 


-4Hhi 

Wvvy, 


FIG.  219. 

drops  to  zero.  The  corresponding  grid  potential  is  measured 
with  the  voltmeter  V.X  This  obtains  when  the  current  registered 
by  the  galvanometer  or  microammeter  in  the  output  circuit  of  the 
amplifier  tube  is  the  value  given  by  this  tube  for  zero  potential 
on  its  grid.  This  can  easily  be  tested  by  short-circuiting  its  input. 
The  input  voltage  is  then  applied  and  the  contact  on  the  poten- 
tiometer again  adjusted  until  the  galvanometer  G  shows  no  change 
in  the  current.  This  grid  potential  is  again  indicated  by  the 
voltmeter  V.  The  difference  between  this  and  the  first  reading 
gives  the  peak  value  of  the  input  voltage. 

It  will  be  noticed  that  the  source  of  a-c.  voltage  which  is  to  be 
measured  is  connected  to  a  circuit  which  can  be  regarded  as  prac- 
tically an  open  circuit.  The  power  consumption,  due  to  the 
electrostatic  capacities  between  the  electrodes  of  the  tube  is  usually 


MISCELLANEOUS  APPLICATIONS  OF  THERMIONIC  TUBES    369 


very  small,  so  that  this  arrangement  acts  practically  as  an  elec- 
trostatic voltmeter,  and  because  of  the  amplification  produced  by 
the  tube,  the  arrangement  can  be  used  to  measure  very  small  alter- 
nating voltages.  Sometimes,  however,  the  plate  current  does 
not  give  a  sharp  intercept  on  the  voltage  axis,  but  tails  off  grad- 
ually, and  the  lower  part  of  the  curve  may  be  very  nearly  linear. 
This  is  shown  in  an  exaggerated  way  in  Fig.  220.  In  such  case 
it  is  best  to  operate  the  tube  a  small  distance  up  the  curve  so  that 


G 

A                       I 

£    \  »| 

1       »p  ]r 

1     

\C           A 

en    ic       A 

B\ 

Orid  Potential 

\ 

-"-:::''J 

\ 

K.-V.V.1 

i    >-!..._ 

* 

i 

i    i 

i 

FIG.  220. 

the  increase  in  current  during  the  positive  half  wave  is  greater 
than  the  decrease  during  the  negative  half  wave.  When  the 
characteristic  tails  off  and  accuracy  is  desired,  it  may  be  necessary 
to  apply  a  correction.  Fig.  221  shows  the  relation  between  the 
true  voltage  and  the  voltage  measured  by  the  tube  which  did  not 
show  a  sharp  intercept  on  the  grid  voltage  axis. 

115.  High  Tension  Voltmeter.  The  three-electrode  tube  makes 
it  possible  to  measure  extremely  high  voltages  with  comparative 
ease,  by  an  arrangement  that  has  been  suggested  to  me  by  Dr. 
E.  R.  Stoekle.  In  this  case  the  high  voltage  to  be  measured  is 


370 


THERMIONIC  VACUUM  TUBE 


applied  between  filament  and  plate.     By  means  D£  a  battery  and 
potentiometer  the  grid  is  adjusted  until  the  current  in  the  plate 


1.0 

1.6 
14 

, 

/ 

/ 

/ 

/ 

/ 

1 

0.4 
0 

/ 

/ 

, 

' 

0.4               0.8                \.Z                 1.6                Z.O              IA 

True  Voltage 
FiG.  221. 


circuit  is  reduced  to  zero.     Since  the  current  through  the  tube  is 
given  by 


it  follows  that  the  voltage  to  be  measured  is  Eg  when  the  current 
is  just  reduced  to  zero.     By  using  a  tube  which  has  a  large  value  of 


MISCELLANEOUS  APPLICATIONS  OF  THERMIONIC  TUBES    371 


/z  the  necessary  grid  battery  voltage  need  not  be  large.  Thus,  if 
ju  =  1000,  a  grid  battery  of  50  volts  is  sufficient  to  measure  50,000 
volts.  Such  a  tube  should  be  designed  with  long  side  tubes  as 
shown  in  Fig.  222,  to  prevent  arcing  across  the  glass. 


116.  The  Audion  Voltage  and  Current  Regulator.  The  follow- 
ing circuit  arrangements,  for  which  I  am  indebted  to  Dr.  P.  I. 
Wold,  indicate  how  the  tube  can  be  used  to  control  the  output  of  a 
generator  or  other  source.  The  generator  S  supplies  power  to 


Line 


Line 


FIG.  223. 


the  line  as  shown  in  Fig.  223.  To  prevent  voltage  fluctuations 
the  vacuum  tube  is  connected  in  series  with  the  field  winding 
of  the  generator.  The  grid  of  the  tube  is  connected  to  a  con- 
venient point  on  the  resistance  R  in  parallel  with  the  field  winding. 
Suppose,  now,  that  the  voltage  of  the  generator  tends  to  increase. 


372 


THERMIONIC  VACUUM  TUBE 


This  increases  the  flow  of  current  through  the  resistance  R,  which 
makes  the  grid  more  negative  than  before,  thus  increasing  the 
resistance  of  the  tube  and,  therefore,  decreasing  the  current 
through  the  tube  and  field  winding,  because  the  tube  and  field 


FIG.  224. 

winding  are  in  series.  By  this  means,  any  tendency  for  the  voltage 
to  increase  is  counteracted.  The  same  thing,  of  course,  holds 
when  the  voltage  tends  to  drop. 

Fig.  224  shows  an  adaptation  of  this  method  of  control  for 
keeping  the  current  output  of  the  generator  constant.     It  will  be 


FIG.  225. 


seen  that  if  the  current  through  the  resistance  R  tends  to  increase, 
the  grid  tends  to  become  more  negative  with  respect  to  the  fila- 
ment, thus  increasing  the  resistance  of  the  tube  and  decreasing 
the  current  through  the  field  winding. 

Fig.  225  shows  a  circuit  arrangement  in  which  the  tube  and 
field  winding  are  connected  in  parallel.  The  high  resistance  R 
to  which  the  grid  is  connected  is  also  in  parallel  with  the  field 
winding.  In  this  case,  when  the  output  voltage  of  the  generator 


MISCELLANEOUS  APPLICATIONS  OF  THERMIONIC  TUBES    373 

tends  to  increase,  the  grid  tends  to  become  less  negative  with 
respect  tc  the  filament,  on  account  of  the  increased  voltage  drop 
established  in  the  resistance  R.  This  tends  to  reduce  the  resistance 
of  the  tube,  thereby  decreasing  the  current  through  the  field  wind- 
ing, which  is  in  parallel  with  the  tube. 

117.  Power-limiting  Devices.  It  has  been  pointed  out  in 
Chapter  VI  that  the  thermionic  valve  acts  not  only  as  a  rectifier 
but  also  as  a  power-limiting  device,  because,  while  it  blocks  current 
in  one  direction,  the  current  in  the  other  direction  cannot  exceed 
the  saturation  value.  This  is,  therefore,  the  maximum  current 
that  can  be  transmitted  through  the  tube. 


J&O.Z 
o. 


Grid  Potential 
FIG.  226. 


The  three-electrode  tube  can  be  very  well  adapted  for  the  pur- 
pose of  limiting  the  output  power.  If,  for  example,  the  grid 
becomes  sufficiently  negative,  the  plate  current  is  reduced  to  zero. 
If,  on  the  other  hand,  the  grid  becomes  sufficiently  positive  the 
plate  current  reaches  a  saturation  value.  It  will  be  evident  from 
the  explanations  given  in  Chapter  VII,  that  this  saturation  value 
may  be  due  to  two  causes:  Firstly,  the  strength  of  the  field  in  the 
space  between  filament  and  grid  may  become  sufficiently  great  to 
pull  the  electrons  away  from  the  filament  as  fast  as  they  are  emitted. 
This  gives  rise  to  the  ordinary  saturation  current.  If  the  cathode 
has  a  smooth  and  pure  surface,  the  knee  of  the  curve,  where  it 
bends  over  to  the  saturation  current,  is  fairly  well  defined.  In 
cases  of  filaments  having  rough  surfaces,  on  the  other  hand,  the 
saturation  current  is  approached  gradually  and  the  curve  does  not 
become  quite  parallel  to  the  voltage  axis.  Under  such  conditions 


374 


THERMIONIC  VACUUM  TUBE 


the  device  is  not  very  suitable  for  power-limiting  purposes,  but 
but  use  can  be  made  of  a  second  factor  which  limits  the  current. 
If  the  external  circuit  contains  a  resistance,  the  potential  of  the 
plate  decreases  as  the  potential  of  the  grid  increases,  on  account  of 
the  voltage  drop  established  in  the  external  resistance  in  the  plate 
circuit.  A  condition  can  then  be  reached  where  the  positive 
potential  of  the  grid  becomes  comparable  with  that  of  the  plate, 
and  in  that  case  a  large  proportion  of  the  electrons  will  be  attracted 
to  the  grid,  thus  limiting  the  flow  of  electrons  to  the  plate.  This 
factor  results  in  a  very  good  curve,  a  sample  of  which  is  shown  in 
Fig.  226.  In  obtaining  this  curve  the  voltages  were  so  adjusted 


FIG.  227. 

that  the  saturation  value  was  obtained  at  a  positive  grid  potential 
equal  to  the  negative  grid  potential  that  just  sufficed  to  reduce  the 
current  to  zero,  thus  resulting  in  a  curve  which  is  nearly  sym- 
metrical with  respect  to  the  axis  of  zero  grid  potential. 

Instead  of  using  a  single  tube  we  can  make  use  of  the  push- 
pull  arrangement  which  was  described  in  Chapter  VII.  This 
gives  a  good  circuit  for  power-limiting  purposes.  The  arrange- 
ment is  shown  in  Fig.  227.  The  input  voltage  was  measured  by 
means  of  a  thermo-couple,  G  and  the  output  current  was  measured 
with  the  a-c.  ammeter  A.  The  result  obtained  is  shown  in  Fig. 
228.  For  low  input  voltages  the  alternating  current  in  the  output 
increased  practically  linearly  with  the  voltage,  but  became  lim- 
ited to  a  value  of  about  3.6  milliamperes,  beyond  which  it  did  not 
increase,  although  the  input  voltage  was  increased  to  10  volts,  as 
shown  by  the  curve.  The  current  was  actually  measured  for 
input  voltages  up  to  42  volts.  At  the  higher  voltages  the  current 
showed  a  tendency  to  decrease.  • 


MISCELLANEOUS  APPLICATIONS  OF  THERMIONIC  TUBES    375 

118.  .The  lonization  Manometer.  It  was  shown  in  Section  36 
that  if  the  gas  pressure  in  a  tube  is  so  low  that  the  mean  free 
path  of  the  electrons  in  the  gas  is  large  compared  with  the  distance 
between  the  electrodes,  the  pressure  is  proportional  to  the  number 
of  positive  ions  formed  by  collision  of  the  electrons  with  the  resid- 
ual gas  molecules.  This  is  expressed  in  equation  (6),  Chapter  V, 
and  was  verified  by  0.  E.  Buckley,1  and  used  by  him  for  the  con- 
struction of  an  ionization  manometer. 


40»Fo 


3.2 


0.5 


4  5 

A.C.1nput-Volt5 

FIG.  228. 


This  device  consists  of  throe  electrodes  connected  in  a  circuit 
as  shown  in  Fig.  229.  The  grid  is  maintained  at  a  positive  poten- 
tial with  respect  to  the  filament,  and  the  plate  is  kept  negative 
with  respect  to  the  negative  end  of  the  filament.  The  electrons 
emitted  from  the  filament  are  attracted  toward  the  grid,  some 
going  to  the  grid  and  some  passing  through  the  openings  in  the 
grid.  Between  the  grid  and  the,  plate,  however,  is  a  retarding 
field,  and,  since  the  plate  is  negative  with  respect  to  the  filament 
none  of  the  electrons  that  are  emitted  from  the  filament  can  reach 
the  plate;  they  attain  their  maximum  speed  in  the  neighborhood 
of  the  grid.  Those  that  pass  through  the  grid  are  retarded  and 
finally  return  to  the  grid  before  they  have  a  chance  of  reaching  the 
plate.  It  is  usually  sufficient  if  the  plate  is  maintained  at  a  poten- 


National  Academy  of  Science,  Vol.  2,  p.  683,  1916.  See  also 
DUSHMAN  and  FOUND  for  calibration  of  this  gauge  for  various  gases,  Phys. 
Rev.,  Febr.,  1920,  p.  133. 


376 


THERMIONIC  VACUUM  TUBE 


tial  of  about  10  volts  negative  with  respect  to  the  negative  end  of 
the  filament.  Now,  if  there  are  gas  molecules  in  the  space,  positive 
ions  will  be  formed  by  collision,  if  the  electrons  move  with  a  suf- 
ficiently high  speed.  The  positive  ions  that  are  formed  between 
filament  and  grid  go  to  the  filament,  but  those  that  are  formed 
between  grid  and  plate  are  attracted  to  the  negative  plate,  thus 
giving  a  current  through  the  galvanometer  G;  and  this  current 
is  a  measure  of  the  number  of  positive  ions  formed.  The  total 
number  of  electrons  flowing  to  the  grid  can  be  measured  with  the 
ammeter  A,  and  must  be  kept  constant.  It  is,  therefore,  desirable 
to  work  on  the  saturation  part  of  the  curve.  The  grid  battery  Ec 


FIG.  229. 


should  be  about  200  volts,  but  depends,  of  course,  on  the  con- 
struction of  the  device. 

This  type  of  gauge  has  the  disadvantage  that  gases  affect  the 
emission  of  electrons  from  the  filament;  that  is,  the  saturation 
current  is  dependent  on  the  amount  and  kind  of  gas  coming  in 
contact  with  the  surface  of  the  filament.  The  electron  emission 
from  oxide-coated  filaments  is  not  as  susceptible  to  the  influence  of 
gas  as  the  emission  from  some  metallic  filaments,  such  as  tungsten, 
and  can  therefore  be  used  to  advantage  in  ionization  manometers. 
If  the  filament  is  operated  at  a  high  temperature,  the  effect  of  gas 
on  the  emission  becomes  less.  (See  Chapter  V.)  Other  means 
can  be  used  for  keeping  the  grid  current  constant.  The  grid  cir- 


MISCELLANEOUS  APPLICATIONS  OF  THERMIONIC  TUBES    377 

cuit  can,  for  example,  be  connected  to  a  regulating  scheme,  similar 
to  those  described  in  Section  116. 

The  ionization  gauge  is  very  convenient  for  measuring  quick 
changes  in  pressure  because  the  positive  current  indicated  by 
the  galvanometer  G  is  a  direct  measure  of  the  amount  of  gas 
present  in  the  device.  On  the  other  hand,  the  manometer  must 
be  calibrated  separately  for  each  kind  of  gas  that  may  be  encoun- 
tered, since  the  amount  of  ionization  produced  depends  on  the 
size  of  the  gas  molecules. 

The  proportionality  between  the  gas  pressure  and  the  positive 
current,  as  shown  by  equation  6,  Chapter  V.,  holds,  in  gauges 
of  the  most  commonly  used  types,  for  pressures  below  about 
1  micron.  The  manometer  can,  therefore,  be  calibrated  against 
a  McLeod  gauge  for  pressures  of  about  1  micron,  or  somewhat  less, 
where  the  McLeod  gauge  gives  reliable  readings.  This,  then 
gives  the  manometer  constant  K  in  the  equation. 


(1) 


where  IP  is  the  positive  current  registered  by  the  galvanometer  G, 
In  the  electron  current  to  the  grid,  and  P  the  pressure  of  the 
gas.  This  equation  can  then  be  used  to  measure  pressures  down  to 
very  low  values  where  the  McLeod  gauge  is  quite  unreliable. 

119.  Heterodyne  Method  of  Generating  Currents  of  Very  Low 
Frequency  with  the  Vacuum  Tube.  In  Section  96  a  circuit  was 
shown  for  the  production  of  low  frequency  currents  with  the  vac- 
uum tube.  The  frequency  is  determined  by  the  constants  of 
the  oscillation  circuit;  hence,  a  very  low  frequency  requires  the 
use  of  large  inductances  and  capacities.  If  it  is  necessary  to 
avoid  the  use  of  such  large  coils  and  condensers,  use  can  be  made 
of  a  scheme  which  was  suggested  to  me  by  Dr.  P.  I.  Wold.  This 
system  consists  of  using  two  vacuum  tube  generators  to  give  fre- 
quencies differing  by  an  amount  equal  to  the  frequency  that 
it  is  desired  to  obtain.  The  output  currents  from  these  tubes  are 
both  impressed  on  the  input  of  another  tube,  which  operates 
as  a  modulator.  The  output  of  the  modulator  contains,  among 
others,  a  frequency  equal  to  the  difference  between  the  frequencies 
impressed  on  its  input.  (See  Section  109.)  Thus,  if  the  two  gene- 
rators give  frequencies  of,  say,  99  and  100  cycles  per  second,  then 
the  output  of  the  modulator  will  contain  a  current  having  a  fre- 


378  THERMIONIC  VACUUM  TUBE 

quency  of  one  cycle  per  second.  In  the  output  of  the  modulator 
can  be  inserted  a  filter  to  by -pass  all  frequencies  higher  than  the 
one  desired.  This  method,  of  course,  requires  that  the  frequencies 
of  the  two  generators  be  maintained  constant  to  a  high  degree, 
since  a  small  change  in  either  of  them  will  cause  a  relatively  large 
change  in  the  low  frequency  obtained  in  the  output  of  the  modu- 
lator. 

120.  The  Thermionic  Valve  as  a  High-  Tension  Switch.     On 
very  high  voltage  power  transmission  lines,  it  is  necessary  to  use 
especially  designed  switches  for  making  and  breaking  the  circuit. 
To  prevent  the  arcing  that  ordinarily  would  take  place  when 
breaking  a  high  voltage  circuit,  the  thermionic  valve  could  be 
used  hi  the  manner  suggested  by  Mr.  J.  R.  Carson.     The  valve 
is  inserted  directly  in  the  line  and  will  transmit  current  in  one 
direction  when  the  filament  is  hot.     When  it  is  necessary  to  stop 
the  flow  of  current,  we  can,  instead  of  directly  breaking  the  cir- 
cuit, simply  cut  out  the  filament  current  of  the  valve.     The  cur- 
rent flowing  through  the  valve  then  dies  down  smoothly  in  a  period 
which  is  short  enough  for  ordinary  work,  but  still  large  enough  to 
prevent  arcing.     For  the  transmission  of  current  in  both  directions, 
we  can,  of  course,  insert  two  valves,  one  to  transmit  current  in 
one  direction  and  the  other  in  the  opposite  direction. 

121.  Devices  Employing  Secondary  Electron  Emission.    The 
emission  of  electrons  from  cold  electrodes  under  the  impact  of 
electrons  (a  phenomenon  which  is  known  as  secondary  electron 
emission  or  delta  rays)  results  in  a  falling  characteristic,  as  shown 
by  the  portion  ABC  of  Fig.  16,  page  48.     The  manner  in  which 
this  characteristic  is  obtained  is  explained  in  Section  23.     A.  W. 
Hull 1  has  made  use  of  this  phenomenon  in  the  construction  of  a 
negative  resistance  amplifier  and  oscillator  and  has  called  the 
device  a  Dynatron.     In  the  circuit  shown  in  Fig.  15,  the  electrons 
coming  from  the  filament  impinge  on  the  grid  and  so  emit  secondary 
electrons  from  it.     These  are  drawn  to  the  plate  by  the  positive 
potential  on  the  plate  supplied  by  the  battery  E.     When  the 
number  of  secondary  electrons  emitted  from  the  grid  becomes  large 
enough  in  proportion  to  the  number  of  electrons  striking  it,  the 
electron  current  flowing  into  the  grid  decreases  as  the  potential 
of  the  grid  is  increased. 

There  is  another  way  in  which  the  tube  can  be  used  to  give  a 
1  Proc.  I.R.E.,  Vol.  6,  p.  5-35,  1918. 


MISCELLANEOUS  APPLICATIONS  OF  THERMIONIC  TUBES    379 

negative  resistance.  In  this  case  the  grid  is  maintained  at  a  posi- 
tive potential  with  respect  to  the  plate.  The  electrons  passing 
through  the  grid  and  striking  the  plate  cause  the  emission  of 
secondary  electrons  from  the  plate  and  these  are  then  drawn 
over  to  the  positive  grid.  The  circuit  arrangement  for  this  case  is 
shown  in  Fig.  230.  If  TO  is  a  resistance  placed  in  the  plate  circuit, 
then  a  potential  Ep  on  the  plate  is  given  by 

77T      77F  7"  /O\ 


where  Ip  is  the  current  in  the  plate  circuit,  and  E»  the  voltage 
impressed  on  the  plate  circuit.     Taking  the  point  B  of  Fig.  16  as 


FIG.  230. 


the  origin  of  coordinates,  the  sloping  part  ABC  of  the  character- 
istic can  be  represented  by  the  equation 


/„=- 


(3) 


where  rp  is  the  plate  resistance  of  the  tube.     Substituting  equation 
(2)  into  this  equation  we  obtain, 


(4) 


Differentiating  Ip  with  respect  to  Eb,  and  multiplying  by  7*0,  we 
get: 

-^_ (5) 

r0-rp 


380  THERMIONIC  VACUUM  TUBE 

that  is,  a  potential  variation  applied  to  the  plate  gives  a  voltage 
variation  in  the  resistance  ro  which  can  be  made  very  large  by 
making  ro  nearly  equal  to  rp.  The  device  thus  operates  as  an 
amplifier. 

Instead  of  using  only  the  negative  resistance  characteristic 
of  the  device,  connected  in  the  manner  explained  above,  Hull  also 
made  use  of  the  normal  amplifying  property  of  the  tube  by  insert- 
ing a  second  grid.  This  device  he  called  a  "  Pliodynatron."  By 
this  means  he  has  been  able  to  obtain  a  voltage  amplification  of 
1000  fold.  To  obtain  such  a  high  voltage  amplification,  however, 
it  is  necessary  to  make  rp  and  ro  nearly  equal.  When  this  is  done 
the  device  becomes  unstable  and  needs  careful  adjustment  and 
constant  attention.  It  was,  however,  found  possible  to  obtain  a 
voltage  amplification  of  100  fold  without  trouble.  It  is  doubtful 
if  a  device  of  this  kind  is  as  good  as  the  audion,  because  by  properly 
designing  the  audion  it  is  easy  to  obtain  a  voltage  amplification  of 
several  hundred  fold,  and  since  the  audion  does  not  possess  a 
falling  characteristic,  its  operation  is  stable  no  matter  how  high 
the  amplification  constant  be  made.  In  cases  where  it  may  be 
necessary  to  use  a  negative  resistance  device,  however,  the  dyna- 
tron  will  be  found  to  be  of  value  and  better  than  an  arc,  which 
also  shows  a  negative  resistance  characteristic.  The  dy natron, 
for  example,  does  not  depend  for  its  operation  on  the  ionization  of 
gas  or  vapor  and  is  therefore  more  reliable. 

The  dynatron  can  be  used  also  to  produce  sustained  oscilla- 
tions if  it  be  connected  in  a  circuit  of  the  type  commonly  used  in 
connection  with  arcs.  Fig.  231,  for  example,  shows  a  circuit 
which  makes  possible  the  production  of  sustained  oscillations 
with  the  dynatron.  It  was  shown  in  Chapter  VIII  that  the 
total  resistance  in  the  output  circuit  will  be  zero  if  the  effective 

resistance  of  the  oscillation  circuit  L;  C,  r,  namely  ro=^-,  is  equal 

Or 

to  the  negative  resistance  of  the  tube.  Hence,  by  adjusting  the 
capacity  C  and  the  inductance  L  so  that  r0  becomes  equal  to  the 
negative  plate  resistance  of  the  tube,  the  total  resistance  of  the 
output  circuit  is  zero,  and  the  device  will  produce  sustained  oscil- 
lations. 

122.  Tubes  Containing  More  than  One  Grid.  Various  investi- 
gators have  suggested  using  two  grids  instead  of  one  for  special 
purposes.  Thus,  R.  A.  Heising,  for  example,  used  a  double  grid 


MISCELLANEOUS  APPLICATIONS  OF  THERMIONIC  TUBES    381 

tube  as  a  modulator,  in  which  the  radio  frequency  is  impressed 
between  the  filament  and  one  grid,  and  the  audio  frequency 
between  the  filament  and  the  other  grid. 


FIG.  231. 

There  are  various  circuits  in  which  the  tubes  with  double  grids 
can  be  used.  In  one  type  of  circuit,  for  example,  the  grid  nearest 
to  the  anode  is  used  as  an  auxiliary  anode,  while  the  grid  nearest 
to  the  cathode  operates  as  a  controlling  electrode.  In  this  case, 


FIG.  232. 


we  can  obtain  the  expression  for  the  effective  voltage  as  follows: 
Referring  to  Fig.  232,  let  us  first  regard  grid  Gi  as  a  cathode.     Let 

the  amplification  constant  of  the  system  so  formed  be  1*2  =  —  ,* 

72 

then  the  effective  voltage  between  Gi  and  G?2  is 


(6) 


where  E'P  is  the  potential  on  the  plate,  and  EC2  the  potential 
on  the  grid  Gi.  This  expression  can  be  regarded  as  the  effective 
anode  potential  for  the  system:  F  (cathode),  G\  (controlling  elec- 


382  THERMIONIC  VACUUM  TUBE 


trode)  and  (^^(anode)  .     Hence,  the  effective  voltage  between 
F  and  G\  is 

.     .     .~v  V  .     .     (7) 


where  Eg\  is  the  potential  on  the  grid  Gi.     Substituting  the  value 
of  Ez  given  by  equation  (6),  we  get  the  effective  voltage  between 


......     (8) 

and  the  current  is 

/2=/2(7i72#'p+7i#*2+^i).  '•&&*   .     •      (9) 

In  the  case  of  a  tube  containing  only  one  grid  the  corresponding 
expression  for  the  current  is 

Ii=fi(-YEP+Ea),    .    .     .  VY    .     .     .     (10) 

where  7  is  the  reciprocal  of  the  amplification  constant  ju. 

If  we  make  the  potential  of  the  plate  in  the  case-  of  the  double 
grid  tube  equal  to  n  times  that  of  the  grid  G%,  we  can  write  equa- 
tion (9): 


/2=/2  [^7172  +  ^)^  +  ^1  J. 


(11) 


This  expression  can  be  compared  with  that  which  holds  for  a  single 
grid  tube.  Suppose  that  the  amplification  constant  of  the  single 
grid  tube  is  such  that  7  =  7172.  Then  we  find  that  the  ratio  of 
the  negative  potentials  on  the  controlling  grid,  that  are  necessary 
to  reduce  the  current  to  zero  in  the  two  cases,  is 


Thus,  suppose  the  potential  of  the  plate  is  the  same  in  both  cases, 
and  let  n  —  2,  and  72  =  0.!.  Then  the  intercept  of  the  character- 
istic on  the  axis  of  controlling  grid  voltage,  in  the  case  of  the  two 
grids,  is  about  six  times  as  large  as  in  the  case  of  the  tube  containing 
only  one  grid.  Other  comparisons  can  be  made  if  the  form  of 
the  characteristic  of  the  double  grid  tube  is  known. 

In  operating  a  double  grid  tube  in  the  manner  described  above, 
the  grid  G%  is  usually  sufficiently  positive  to  draw  an  appreciable 

1  See  also  BARKHAUSEN,  Jahrb.  d.  drahtlosen  Tel.  u.  Tel,  Vol.  14,  p.  43, 
1919. 


MISCELLANEOUS  APPLICATIONS  OF  THERMIONIC  TUBES    383 

number  of  electrons  to  it,  and  this  decreases  the  current  to  the 
plate.  In  general,  such  tubes  are  not  as  good  as  audions  when 
used  for  ordinary  purposes.  Circuits  can,  of  course,  be  easily 
devised  which  enable  one  to  make  use  of  variations  in  the  current 
flowing  to  the  second  grid,  as  well  as  that  flowing  to  the  plate  or 
anode. 

Another  way  in  which  double  grid  tubes  can  be  used  is  to 
use  the  grid  nearest  to  the  anode  as  the  controlling  electrode,  and 
apply  a  positive  potential  to  the  grid  nearest  to  the  cathode, 
as  has,  for  example,  been  done  by  Langmuir.  This  grid  should 
then  preferably  be  placed  close  to  the  cathode,  and  the  potential 
applied  to  it  should  be  high  enough  to  pull  the  electrons  away 
from  the  filament  as  fast  as  they  are  emitted  from  it,  thus  giving 
the  condition  for  the  saturation  current.  In  this  case  the 
space  charge  between  the  filament  and  the  first  grid  is  small. 
If  the  second  grid  is  kept  negative  with  respect  to  the  first 
grid,  the  electrons  passing  through  the  first  grid  will  be  slowed 
down  in  approaching  the  second  grid,  thus  increasing  the  space 
charge  between  the  two  grids,  but  the  electrons  in  the  space  are 
now  spread  throughout  a  greater  volume,  instead  of  being  con- 
centrated around  the  filament,  and  hence,  potential  variations 
applied  to  the  second  grid  can  be  expected  to  produce  relatively 
large  changes  in  current  flowing  to  the  anode.  Here,  again,  the 
first  grids  robs  the  plate  of  current,  but  the  circuit  could  be  so 
arranged  that  use  is  made  of  the  variation  in  both  currents,  namely, 
that  flowing  to  the  plate  and  that  to  the  first  grid. 

In  using  such  devices,  where  electrons  impinge  on  a  conductor 
which  does  not  have  the  highest  positive  potential  in  the  system, 
the  effect  of  secondary  electron  emission  must  be  taken  into  con- 
sideration, because  if  the  electrons  impinge  with  sufficient  violence 
on  a  conductor,  secondary  electrons  are  emitted  from  it,  and  if 
there  is  another  conductor,  which  is  positive  with  respect  to  the 
first,  the  secondary  electrons  will  be  drawn  over  to  this  positive 
conductor.  If  the  velocity  with  which  the  electrons  impinge  on 
the  first  conductor  is  large  enough,  so  many  secondary  electrons 
can  be  emitted  that  electrons  will  flow  out  of  this  conductor 
instead  of  into  it,  thus  reversing  the  direction  of  the  current. 


INDEX 


Abraham,  2,  8,  227 
Admittance,  mutual,  187 

of  plate  circuit,  280 
Aeroplane,  radio  transmitter,  240 

radio  receiver,  240 
Amplification: 

circuits,  249  et  seq. 

distortionless,  260 

equations,  180  et  seq. 
verification  of,  189 

expressed  on  logarithmic  scale,  218 

expressed    in    miles    of    standard 
cable,  218 

as  function  of  operating  parame- 
ters, 224 

measurement  of,  215 

power,  185 

voltage,  181,  213 
Amplification  constant,  160 

calculation  of,  227 

equation  for,  231 

for  cylindrical  structures,  234 

measurement  of,  203,  215 

methods  of  measuring,  193 
Amplifier,  circuit  with  single  source 
of  voltage,  262 

multi-stage,  252 

push-pull,  261 

telephone,  262 

types  of,  236,  249  et  seq. 

voltage,  252 

unilateral,  262 

Amplitude  of  oscillations  in  three- 
electrode  tube,  279 
Amplitude  factor,  125 

values  of,  128 


Anode : 

power  dissipated  at,  75,  303 

effects  of  temperature  of,  76 
Anode  potential: 

effect  of,  on  power  from  oscillator, 
312 

effect  of,  on  oscillation  current,  311 

effect  of,  on  amplification,  226 
Anode  temperature: 

effects  of,  in  vacuum  tubes,  76 
Appleton,  E.  V.,  202 
Applications: 

miscellaneous,  of  thermionic  tubes, 

367 
Arc  discharge: 

difference  between    gas-free    and, 

107 

Armstrong,  E.  H.,  290,  335 
Arnold,  H.  D.,  81,  170,  188,  253,  255 
Artificial  line,  216 
Attenuation  constant,  217 

of  standard  cable,  218 
Audibility : 

expressed  in  miles  of  cable,  351 
Audibility  method  of  measuring  de- 
tecting current,  336,  349 
Audion,  145 

von  Baeyer,  42 
Ballantine,  S.,  199,  285 
Barkhausen,  155,  382 
Blue  glow,  21 
Bohr,  18 
Boltzmann,  25 
Bridge: 
use  of  audion  in,  213 


385 


386 


INDEX 


Bright  spots  of  coated  cathodes,  85 
Buckley,  O.  E.,  375 

Campbell,  G.  A.,  139 
Capacity: 

input,  208,  212 

electrode,  see  electrode,  205 
Carbon  button  generator,  223 
Carrier  wave,  318 
Carson,  J.  R.,  261,  316,  378 
Cathode: 

equipotential,  55 

efficiency,  76 
Characteristic: 

for  concentric  cylinders,  59 

effect  of  curvature  of,  on  operation 
of  tube,  73 

dynamic,  295 

effect  of  resistance  on,  170 

grid  current,  186 

lumped,  160 

of  inductive  plate  circuit,  174 

of  non-inductive  plate  circuit,  169 

grid  current-grid  potential,  153 

plate  current-grid  potential,  160 

plate  current-plate  potential,  150 

plate  resistance,  196 

static  and  dynamic,  170  et  seq. 

verification  for  thermionic  ampli- 
fier, 158 

of  thermionic  valve,  50 

for  parallel  plates,  54 

influence  of  initial  velocities  on,  61 

straightened  by  resistance,  128 
Child,  C.  D.,  55,  58 
Coated  filament,  81 

operating  temperature  of,  81 

See  also  Wehnelt  cathode. 
Collision,  elastic,  20 

ionization  by,  21 
Colpitts,  E.  H.,  261,  294,  306 
Colpitts'  oscillator  circuit,  282 
Comparison  of  detectors,  346 
Complex  circuits,  290 
Conductance: 

mutual,  166,  189,  269 

measurement  of  mutual,  199,  203 

reflex  mutual,  166 


Conductance — Continued 

relation  between  mutual,  of  tube 

and  circuit,  279 
Contact  potential  difference,  26 

measurement  of,  28 
Convergence  frequency,  21 
Corpuscle,  1 
Coupled  circuits,  290 
Coupling: 

method  of  adjusting,  306 
Current  limitation : 

by  space  charge,  52 

by  voltage  drop  in  filament,  64,  69 

by  thermionic  emission,  70 
Curvature  of  characteristic: 

effect  of,  on  operation  of  tube,  73 

Davisson,  C.  J.,  80,  82 

Debije,  P.,  26 

Delta  rays,  47 

Dember,  H.,  47 

Detecting  current,  169,  325 

method  of  measuring,  335 

r.m.s.  value  of,  328 

as  function  of  anode  potential  with- 
out grid  condenser,  331 

with  grid  condenser,  359 
Detecting  efficiency,  34.4 
Detection : 

double,  365 

with   blacking   condenser   in   grid 
circuit,  332 

theory  of,  315 
Detection  coefficient,  326 

measurement  of,  339 

as  function  of  plate  and  grid  poten- 
tials, 328 

Detectors,  comparison  of,  346 
Dislodged  electrons,  current  carried 

by,  57 
Dislodgment  of  electrons: 

means  of,  17,  30 

from  curved  surfaces,  35 
Distortion : 

due  to  harmonics,  168 

reduced  by  external  resistance,  169 
Distortionless  amplification,  178 
Dushman,  77,  84,  122,  129,  375 


INDEX 


387 


Dynatron,  378,  380 

Eccles,  155 
Edison  effect,  30 
Efficiency : 

of  cathode,  76  et  seq. 

thermionic,  77 
for  tungsten,  77 
for  coated  filament,  80 

detecting,  344 

of  thermionic  oscillator,  298 

rectification,  123  et  seq. 
Einstein,  20,  41 
Electrode  capacities,  178,  205 
Electron : 

accelerated,  13 

current,  57 

dislodgment  of,  14 

effect  of  electric  field  on  motion  of, 
9 

effect  of  magnetic  field  on  motion 
of,  10 

electromagnetic  mass  of,  7 

field  of  moving,  5 

field  of  "stationary,"  4 

free,  23 

longitudinal  mass  of,  8 

mass  of,  2,  6,  8 

size  of,  2,  8 

transverse  mass  of,  8 
Electrons : 

free,  16 

dislodgment  of,  17,  30,  35 

in  equilibrium,  25 

occurrence  of,  16 
Electron  affinity,  24 

effect  of  surface  condition  of  cath- 
ode on,  34 

effect  of,  on  saturation  current,  79 

relation  between  Richardson's  con- 
stant b  and,  33 

values  of,  29 

Electron  evaporation  constant,  24 
Electrostatic  voltmeter: 

vacuum  tube  as,  367 
Elster  and  Geitel,  30 
Emissivity,  thermal,  75 
Epstein,  P.  S.,  63 


Equipotential  cathode,  55 
Everitt,  H.  W.,  200,  203 

Falling  characteristic,  170 
Faraday,  2 
Feed-back : 

Amplifier,  257 

receiving  circuit,  360 
Filament  current: 

effect  of,  on  amplification,  225 

on  oscillation,  308 
Filament  voltage  drop: 

effect  on  characteristic,  64,  69 

effect  on  detecting  current,  329 
Filter,  138,  216,  265,  see  also  Wave 

Filter 

Fleming,  30,  111,  125 
de  Forest,  42,  145 
Form  factor,  125 

values  of,  128 
Found,  C.  G.,  375 
Franck  and  Hertz,  20 
Free  electrons,  23 
Frequency : 

effect  of,  in  reducing  rectification, 
134 

heterodyne  method  of  generating 
low,  377 

obtainable  with  thermionic  oscilla- 
lator,  312 

of    oscillations    in    three-electrode 

tube,  274 
Fry,  T.  C.,  63,  138 

Gas: 

effect  of,  on  discharge,  86 

effect  of,  on  electron  emission,  98 

surface  effect  of,  86,  98 

volume  effect  of,  86 
Gauge,  ionization,  91,  375 
Gherardi,  B.,  262 
Graetz,  130 
Grainacher,  H.,  141 
Grid: 

action  of,  146 

effect  of  dimensions  of,  228 

screening  effect  of,  229 


388 


INDEX 


Grid  current  characteristic,  153 

in  oscillator,  295 
Grid  leak  resistance,  309 
Grid  potential,  means  of  maintaining, 

250 
Grids,  tubes  containing  two,  380 

Hallwacks,  38 
Harmonics,  168,  177,  282 
Hartley,  257,  282,  287,  293,  313 

oscillator  circuit,  282,  313 
Hazeltine,  L.  A.,  165,  269,  280,  283 
Heising,  280,  283,  308,  310,  323,  367, 

380 

Hertz,  H.,  38 
Hertz  and  Franck,  20 
Heterodyne: 

method  of  generating  low  frequen- 
cy, 377 

reception,  353  et  seq. 
High  tension  voltmeter: 

vacuum  tube  as,  369 
Homodyne,  reception,  358 
Hull,  A.  W.,  49,  132,  378,  380 

Impact,  elastic,  20 
Impedance,  input,  206,  212 
Infra-saturation  part  of  characteris- 
tic, 71 
Initial  velocities : 

of  photo-electrons,  38 

influence  of  on  tube  characteristic, 

61 

Input   capacity,   impedance,   power, 
etc.,  see  corresponding  nouns. 
Ion: 

negative,  19 

positive,  18 
lonization,  16 

by  collision,  21 

directive  effect  on  flow  of  gas,  101 

effect  on  infra-saturation,  93 

effect  on  operation  of  oscillator,  101 

effects  of,  91 

gauge,  91,  375 

heating  of  cathode  by,  92 

at  high  pressure,  106 

at  low  pressures,  90 


lonization,  manometer,  375 
proportional  to  low  pressure,  91 
voltage,  21,  22 

Jewett,  F.  B.,  262 
Johnson,  J.  B.,  213 

King,  R.  W.,  227,  234,  235 

Langmuir,  I.,  55,  59,  75,  101,  154, 
245,  383 

Latour,  M.,  153,  187 

von  Laue,  227 

Lenard,  42,  53 

Life  of  a  vacuum  tube,  84 

Lilienfeld,  53 

Limitation  of  current,  52 
by  space  charge,  52 
by  voltage  drop  in  filament,  64,  69 
by  thermionic  emission,  70 

Limiting,  power  device,  373 

Lorentz,  2,  8 

n,  see  Amplification  Constant. 

Manometer,  inozation,  375 

Marconi,  112 

Maxwell,  227,  232 

Mean  free  path  of  electrons  in  gases, 

88  et  seq. 
Meissner,  290 
Microphone  generator,  223,  339 

for  obtaining  modulated  waves,  348 
Miles  of  standard  cable: 

amplification  expressed  in,  218 

audibility  expressed  in,  351 

relation  between  amplification  and, 

219 

Miller,  J.  M.,  194,  197,  205,  209,  290 
Millikan,  R.  A.,  13,  41 
Modulated  current,  r.m.s.  value  of, 

328 
Modulated  wave: 

equation  for,  319 

completely,  321 

method  of  obtaining  completely, 

342     .- 
Modulation : 

double,  365 


INDEX 


389 


Modulation — Continued 

systems  of,  322 

theory  of,  315 
Morecroft.  J.  H.,  303 
Multiplex  telegraphy  and  telephony, 

364 
Multi-stage  amplifier,  184,  252 

design  of,  256 

with  inductive  connection,  252 

with  inter-tube  transformers,  253 

with  non-inductive  connection,  253 

phase  relations  in,  260 
Music,  transmission  of,  254 
Mutual  admittance,  see  Admittance. 
Mutual  conductance,    see  Conduct- 
ance. 

Negative  carriers,  71,  96 

Negative  resistance,  48,  108,  379,  see 

also  Resistance,  271 
Nichols,  205,  215 

Occluded  gases,  effects  of,  102 
Operating  parameters,  influence  of: 

on  amplification,  224 

on  oscillation,  307 
Optimum    voltage   for    rectification, 

117  et  seq. 
Oscillation: 

amplitude  of,  in  vacuum  tube,  279 

conditions  for,  267 

in  two-electrode  tube,  269 
in  three-electrode  tube,  271 
Oscillation    current    as    function    of 
filament  current,  309 

of  anode  potential,  311 

equations  of  vacuum  tube,  267 

generator,  266  et  seq. 
Oscillations  in  amplifier  circuits,  258 
Oscillator: 

tuned  grid  circuit,  284 

with  grid  leak  resistance,  309 

with  grid  battery,  310 

practical,  circuits,  292 

for  extreme  frequencies,  312 
Oscillograms : 

of  current  in  valve  rectifier,  129 

of  oscillator  output,  302,  305 


Parasitic  circuits,  285 
Peltier  effect,  27 
Phase  relations: 

in  amplifier,  174 

in  multi-stage  amplifier,  260 

in  oscillator,  280 
Photo-electric : 

effect,  38 

equation,  41 

long  wave-length  limit,  42 
Photo-electrons,    maximum   velocity 

of,  38 

Planck,  18,  20 
Plate  impedance,  160 
Plate    resistance,    see    also    Resist- 
ance. 

characteristic,  196 
Plate  current,  177 
Pliodynatron,  380 
Poisson,  15,  52,  56 
Positive  electron,  2 
Potential  distribution: 

in  audion,  147,  148 

for  finite  initial  velocities,  62 

for  zero  initial  velocity,  56 
Power: 

effect  of  anode  potential  on,  from 
oscillator,  312 

in  input  circuit,  211 

in    output     circuit    of    amplifier, 
192 

in  output  circuit  of  detector,  328 

in  output  circuit  of  oscillator,  296 
Power  amplification,  185 
Power-limiting  devices,  373 
Pure  electron  discharge,  22 
Push-pull  amplifier,  261 


Radiation: 

from  atoms,  19 

due  to  accelerated  electron,  13 
Radiation  constant,  75 
Radio    transmitting    and    receiving 

systems,  361  et  seq. 
Reactance,  input,  208 
Receiver  shunt,  216 

computation  of,  222 


390 


INDEX 


Receiver  shunt  for  measuring  ampli- 
fication, 218 

for  measuring  audibility,  350 

for   measuring   detecting   current, 
337 

theory  of,  221 

Receiving  systems,  radio,  361 
Recombination  of  ions,  96 
Rectification : 

conditions  for,  109 

efficiency,  123  et  seq. 
Regeneration,  287 
Regulator : 

audion  as  current  and  voltage,  371 

valve  as  voltage,  142 
Repeater: 

Western  Electric  type,  239 

circuit,  263 
Resistance: 

computation  of  plate,  234 

input,  208,  209,  212 

measurement  of  plate,  195 

negative,  271,  379 

plate,  268 

Richards,  W.  L.,  262 
Richardson,  O.  W.,  24,  26,  31,  32,  53, 
83 


Saturation  current,  50,  77 

from  contaminated  tungsten,  37 
effect  of  surface  condition  on,  37 
from  oxide-coated  cathodes,  37 

Schelleng,  J.  C.,  303 

Schottky,  55,  155,  227 

von  Schweidler,  53 

Secondary  electron  emission,  47,  383 
devices  employing,  378 

Singing: 

in  amplifiers,  209,  259 
in  repeater  circuits,  263 

Soddy,  53 

Space  charge,  15,  55 

current  limitation  by,  52 
due  to  positive  and  negative  car- 
riers, 52 

relation  between  potential  distribu- 
tion and,  15 


Space  current,  6,  57 

control  of,  by  third  electrode,  42 

as  function  of  plate  and  grid  poten- 
tials, 46 
Standard  cable: 

constants  of,  218 

Stefan-Boltzmann  radiation  law,  75 
Stevenson,  G.  H.,  198 
Stoller,  H.  M.,  142 
Stoletow,  53 
Stoney,  G.  J.,  1 
Strayfield,  146,  229 

verification  of,  relation,  157 
Structural  parameters  of  tube,  226 
Surface  condition  of  cathode: 

effect  of,  on  electron  affinity,  34 
on  saturation  current,  37 
on  characteristic,  70 

force  on  electrons,  23,  24 
Switch,  valve  as  high  tension,  378 

Telegraphy,  multiplex,  364 
Telephone  amplifier,  262 
Telephony,  multiplex,  364 
Temperature : 

effects  of  anode,  on  operation  of 
tube,  76 

safe  cathode,  85 

saturation,  54,  308 
Thermionic  amplifier: 

types  of,  236  et  seq. 
Thermionic  valve,  50 

characteristics  of,  51 

as  current  limiting  device,  124 

as  high  power  rectifier,  115 

as  high  tension  switch,  378 

minimum  resistance  of,  126 

types  of,  120 

voltage  drop  in,  115 
Thermionic  efficiency,  77 

of  coated  cathode,  80 

effects  of  electron  affinity  on,  78 

of  tungsten,  77 
Third  electrode  for  controlling  space 

current,  42,  146 
Thomson,  J.  J.,  1,  5,  55 
Three-halves  power  equation: 

for  parallel  plates,  58 


INDEX 


391 


Three-halves  power  equation — Con. 

for  concentric  cylinders,  60 
Transformer,  input  and  output,  250 
Transmitting  systems,  radio,  361 
Two-way,  one-repeater  circuit,  264 
Two-way,  two-repeater  circuit,  263 

Vacuum,  by  vaporization  of  calcium, 

53 

Vacuum  test  for  tubes,  105 
Vacuum  tubes: 

types  of,  120  et  seq.,  236  et  seq. 
Vallauri,  154 
Valve  detector,  111 

with  anode  battery,  112 
See  also  Thermionic  Valve. 
Voltage  amplification,  181,  213,   see 

Amplification. 

Voltage  drop  in  filament,  see  Fila- 
ment, 


Voltmeter : 
vacuum  tube  as  electrostatic,  367 

Wave  filter,  138,  see  also  Filter. 

Wave  shape: 

of  current  in  output  of  audion,  166 
of  current  in  valve,  116 
of  modulated  current,  320 

Wehnelt  cathode,  25,  see  also  Coated 
Filament. 

White,  W.  C.,  313 

Wilson,  H.  A.,  99 

Wilson,  R.  H.,  251,  355,  368 

Wilson,  W.,  64 

Wold,  P.  I.,  371,  377 

X-radiation: 

characteristic,  47 
general,  47 
soft,  47 


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